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": "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_005201",
"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 probability must lie between 0 and 1 inclusive."
... | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005202",
"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_005203",
"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): 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_005204",
"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): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005205",
"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_005206",
"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_005207",
"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_005208",
"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: 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_005209",
"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_005210",
"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_005211",
"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: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_005212",
"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: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005213",
"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: A probability must lie between 0 and 1 inclusive."
... | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005214",
"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_005215",
"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: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005216",
"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_005217",
"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_005218",
"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: 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_005219",
"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): 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_005220",
"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: 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_005221",
"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): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005222",
"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): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_005223",
"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 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_005224",
"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: 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_005225",
"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_005226",
"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: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005227",
"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_005228",
"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_005229",
"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): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005230",
"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_005231",
"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): 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_005232",
"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_005233",
"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: 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_005234",
"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: 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_005235",
"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_005236",
"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): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_005237",
"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_005238",
"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": "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_005239",
"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_005240",
"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_005241",
"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_005242",
"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_005243",
"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_005244",
"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: 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_005245",
"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 probability must lie between 0 and 1 inclusive."
... | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005246",
"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_005247",
"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_005248",
"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_005249",
"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: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005250",
"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_005251",
"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: A probability must lie between 0 and 1 inclusive."
... | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005252",
"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_005253",
"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_005254",
"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): 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_005255",
"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_005256",
"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_005257",
"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_005258",
"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_005259",
"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): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005260",
"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): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005261",
"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: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005262",
"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_005263",
"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_005264",
"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: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_005265",
"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): 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_005266",
"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": "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_005267",
"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): 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_005268",
"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): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005269",
"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_005270",
"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: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005271",
"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_005272",
"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_005273",
"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 probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005274",
"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_005275",
"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_005276",
"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_005277",
"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): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005278",
"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_005279",
"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: 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_005280",
"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_005281",
"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): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005282",
"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_005283",
"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): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005284",
"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: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_005285",
"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: 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_005286",
"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): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_005287",
"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): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_005288",
"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_005289",
"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 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_005290",
"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_005291",
"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_005292",
"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": "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_005293",
"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): 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_005294",
"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): 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_005295",
"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): 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_005296",
"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: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_005297",
"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_005298",
"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: 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_005299",
"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_005300",
"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"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.