prompt dict | response string | meta dict | fact_sources list | artifacts dict |
|---|---|---|---|---|
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006301",
"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_006302",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A 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_006303",
"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_006304",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006305",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006306",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistic... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006307",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006308",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stat... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006309",
"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_006310",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "core",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006311",
"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_006312",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006313",
"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_006314",
"domain": "master_scholars_academics",
"subdomain": "stats::probability",
"task": "fact_check",
"difficulty": "advanced",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stati... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006315",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."
}
]
} | TRUE — The standard deviation is the square root of the variance. | {
"id": "WUA_STAT_25K_006316",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statisti... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."
}
]
} | TRUE — Independence means P(A∩B) = P(A)P(B). | {
"id": "WUA_STAT_25K_006317",
"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): Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
"id": "WUA_STAT_25K_006318",
"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"statistics",... | [
{
"source_title": "OpenStax Introductory Statistics (CC BY 4.0)",
"source_url": "https://openstax.org/details/books/introductory-statistics",
"evidence_type": "standard",
"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
"max_tokens_hint": 220,
"style_contract": "concise_formal"
} |
{
"messages": [
{
"role": "system",
"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
},
{
"role": "user",
"content": "True or False (academics): A probability must lie between 0 and 1 inclusive."
}
]
} | TRUE — A probability must lie between 0 and 1 inclusive. | {
"id": "WUA_STAT_25K_006319",
"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_006320",
"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_006321",
"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",
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{
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},
{
"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). | {
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{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
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{
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{
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},
{
"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). | {
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"domain": "master_scholars_academics",
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"messages": [
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},
{
"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. | {
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"domain": "master_scholars_academics",
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"task": "fact_check",
"difficulty": "seed",
"truth_mode": "verifiable_fact",
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"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
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{
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{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
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{
"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). | {
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"domain": "master_scholars_academics",
"subdomain": "stats::probability",
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"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)",
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},
{
"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_006388",
"domain": "master_scholars_academics",
"subdomain": "stats::distribution",
"task": "fact_check",
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"wrapper": "verify_true_false",
"created_utc": "2026-01-04T05:02:58",
"license": "cc-by-4.0",
"tags": [
"stat... | [
{
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},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
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{
"messages": [
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},
{
"role": "user",
"content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."
}
]
} | FALSE — Correlation alone does not establish causation. | {
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"domain": "master_scholars_academics",
"subdomain": "stats::modeling",
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"license": "cc-by-4.0",
"tags": [
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},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
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{
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"content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."
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{
"role": "user",
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} | FALSE — Correlation alone does not establish causation. | {
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"domain": "master_scholars_academics",
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"tags": [
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{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
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{
"messages": [
{
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{
"role": "user",
"content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."
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]
} | FALSE — Correlation alone does not establish causation. | {
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"domain": "master_scholars_academics",
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"license": "cc-by-4.0",
"tags": [
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},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
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"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_006392",
"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... | [
{
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},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
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} |
{
"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_006393",
"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)",
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"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
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"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_006394",
"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)",
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},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
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"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). | {
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"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",
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"accessed_at_utc": "2026-01-04T05:02:58"
},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
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"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_006396",
"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)",
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},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
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{
"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_006397",
"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_006398",
"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",
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},
{
"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods",
"sou... | {
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"style_contract": "concise_formal"
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{
"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_006399",
"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_006400",
"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"
} |
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