prompt dict | response string | meta dict | fact_sources list | artifacts dict |
|---|---|---|---|---|
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006801",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006802",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006803",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006804",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006805",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006806",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006807",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothes... | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006808",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006809",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006810",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006811",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006812",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006813",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006814",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothes... | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006815",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006816",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006817",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006818",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006819",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006820",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006821",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006822",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006823",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A probability must lie between 0 and 1 inclusive."
... | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006824",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006825",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006826",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006827",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A probability must lie between 0 and 1 inclusive."
... | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006828",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A probability must lie between 0 and 1 inclusive."
... | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006829",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006830",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothes... | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006831",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006832",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006833",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006834",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the va... | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006835",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006836",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothes... | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006837",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the va... | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006838",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006839",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006840",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006841",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006842",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006843",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006844",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006845",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006846",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006847",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006848",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006849",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006850",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006851",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006852",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006853",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stat... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006854",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006855",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006856",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006857",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006858",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006859",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006860",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006861",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006862",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006863",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006864",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006865",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the va... | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006866",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006867",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006868",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006869",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006870",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006871",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006872",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stat... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006873",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006874",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006875",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006876",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006877",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stat... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006878",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006879",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the va... | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006880",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stat... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A probability must lie between 0 and 1 inclusive."
... | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006881",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006882",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the va... | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006883",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006884",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006885",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006886",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the va... | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006887",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006888",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006889",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stat... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006890",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006891",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006892",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006893",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothes... | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006894",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006895",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006896",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006897",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006898",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stat... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."
}
]
} | FALSE — A p-value is P(data as or more extreme | null), not P(null | data). | {
"id": "WUA_STAT_25K_006899",
"domain": "master_scholars_academics",
"subdomain": "stats::hypothesis_testing",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"st... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006900",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.