[
  {
    "local_id": "C1",
    "type": "empirical",
    "core": true,
    "statement": "Over every sample size n from 5 to 100 and every true proportion p from 0.001 to 0.999 in steps of 0.001, the exact coverage of the nominal 95% Wald interval for a binomial proportion is below 0.93 at a fraction 0.463088 of the 95904 (n, p) pairs and below 0.95 at a fraction 0.923027 of them.",
    "evidence": [
      {"result": "R1.share_below_low.wald", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.share_below_nominal.wald", "produced_by": "code/coverage.py", "tolerance": 0.000001}
    ],
    "depends_on": [],
    "falsified_if": "An exact computation of the Wald interval's coverage over the same grid finds a fraction below 0.93 outside 0.463087 to 0.463089, or a fraction below 0.95 outside 0.923026 to 0.923028.",
    "confidence": 0.98
  },
  {
    "local_id": "C2",
    "type": "empirical",
    "core": true,
    "statement": "Over the same 95904 pairs, the coverage of the nominal 95% Wilson score interval is below 0.93 at a fraction 0.032981 of them and that of the Agresti-Coull interval at a fraction 0.003681, with mean coverages of 0.952036 and 0.958695 against the Wald interval's 0.88228.",
    "evidence": [
      {"result": "R1.share_below_low.wilson", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.share_below_low.agresti_coull", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.mean_coverage.wilson", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.mean_coverage.agresti_coull", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.mean_coverage.wald", "produced_by": "code/coverage.py", "tolerance": 0.000001}
    ],
    "depends_on": [],
    "falsified_if": "An exact computation over the same grid gives any of these five values more than 0.000001 away from the stated one.",
    "confidence": 0.98
  },
  {
    "local_id": "C3",
    "type": "empirical",
    "core": true,
    "statement": "The nominal 95% Clopper-Pearson interval covers at least 0.95 at every one of the 95904 pairs, with a lowest coverage of 0.9502, and its mean coverage of 0.970942 exceeds the nominal level by about 0.021.",
    "evidence": [
      {"result": "R1.min_coverage.clopper_pearson", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.mean_coverage.clopper_pearson", "produced_by": "code/coverage.py", "tolerance": 0.000001}
    ],
    "depends_on": [],
    "falsified_if": "Some pair on the grid has Clopper-Pearson coverage below 0.95, or the mean coverage over the grid lies outside 0.970941 to 0.970943.",
    "confidence": 0.98
  },
  {
    "local_id": "C4",
    "type": "empirical",
    "core": false,
    "statement": "The Wald interval's coverage at a fixed p can fall sharply with one more observation: at p = 0.2 its largest fall for n from 5 to 100 is from 0.94441 at n = 14 to 0.814815 at n = 15, and at p = 0.05 from 0.946709 at n = 58 to 0.798385 at n = 59.",
    "evidence": [
      {"result": "R1.wald_at_fixed_p.p_0_2.largest_drop.from_n", "produced_by": "code/coverage.py"},
      {"result": "R1.wald_at_fixed_p.p_0_2.largest_drop.from", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.wald_at_fixed_p.p_0_2.largest_drop.to", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.wald_at_fixed_p.p_0_05.largest_drop.from_n", "produced_by": "code/coverage.py"},
      {"result": "R1.wald_at_fixed_p.p_0_05.largest_drop.from", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.wald_at_fixed_p.p_0_05.largest_drop.to", "produced_by": "code/coverage.py", "tolerance": 0.000001}
    ],
    "depends_on": [],
    "falsified_if": "The exact Wald coverage at p = 0.2 for n = 14 and 15, or at p = 0.05 for n = 58 and 59, differs from the stated values by more than 0.000001.",
    "confidence": 0.97
  },
  {
    "local_id": "C5",
    "type": "empirical",
    "core": false,
    "statement": "Even at n = 100, the Wald interval's coverage is 0.633433 at p = 0.01 and 0.877463 at p = 0.05.",
    "evidence": [
      {"result": "R1.wald_at_fixed_p.p_0_01.at_n_max", "produced_by": "code/coverage.py", "tolerance": 0.000001},
      {"result": "R1.wald_at_fixed_p.p_0_05.at_n_max", "produced_by": "code/coverage.py", "tolerance": 0.000001}
    ],
    "depends_on": [],
    "falsified_if": "The exact Wald coverage at n = 100 differs from 0.633433 at p = 0.01, or from 0.877463 at p = 0.05, by more than 0.000001.",
    "confidence": 0.98
  }
]