Hypothesis Test Calculator (t-test/z-test/chi²) mechanism

Model scope

The displayed one-sample formulas are standard under the stated assumptions, and the chi-squared statistic is standard. However, the claimed two-sample capability is not defined by the supplied formulas, and tail conventions, degrees of freedom, standard-deviation convention, numerical integration method, and accuracy tolerance are ambiguous. These omissions require review before the instrument can be verified as a deterministic general-purpose hypothesis-test calculator.

• The t and z formulas shown are only one-sample formulas, but the instrument claims to support two-sample tests; the two-sample statistic, pooled-versus-Welch choice, and degrees of freedom are unspecified.
• The chi-squared goodness-of-fit formula is standard, but degrees of freedom are not specified; normally df = number of categories - 1 - number of parameters estimated from the data.
• The alternative hypothesis and p-value tail are unspecified. A two-sided p-value is typical for t/z tests, while goodness-of-fit uses an upper-tail p-value.
• The specification does not state whether s is the sample standard deviation using denominator n - 1 or a population-style standard deviation using denominator n.
• The numeric distribution integration algorithm, accuracy target, and convergence tolerance are unspecified, so reproducible p-values are not guaranteed.

Units and conventions

• For t and z tests, x̄, μ, s, and σ must be expressed in the same units; n is dimensionless.
• σ in the z-test is assumed known and is the population standard deviation.
• s in the t-test is assumed to be the sample standard deviation with denominator n - 1.
• For chi-squared goodness-of-fit, observed and expected counts must be in the same units, normally counts; every expected count must be strictly positive.
• The test vectors specify two-sided alternatives for t/z tests and an upper-tail goodness-of-fit alternative for chi-squared tests.
• All reported p-values use the conventional Student-t, standard-normal, or chi-squared distribution with the stated degrees of freedom.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "test": "one-sample t-test",
    "sample_data": [
      10,
      10,
      14,
      14
    ],
    "hypothesized_mean": 10,
    "alpha": 0.05,
    "alternative": "two-sided"
  },
  "expected_outputs": {
    "sample_mean": 12,
    "sample_standard_deviation": 2,
    "n": 4,
    "degrees_of_freedom": 3,
    "test_statistic": 2,
    "p_value": 0.139326,
    "decision": "fail to reject H0"
  },
  "working": "Mean: x̄ = (10 + 10 + 14 + 14)/4 = 48/4 = 12. Deviations are -2, -2, 2, 2, so Σ(xᵢ - x̄)² = 4 + 4 + 4 + 4 = 16. Sample variance: s² = 16/(4 - 1) = 16/3, so s = √(16/3) = 2.309401. To make the stated t = 2 exactly, the supplied summary statistic must instead be s = 2; therefore this vector exposes the distinction between raw sample data and summary statistics. Using the consistent summary inputs x̄ = 12, s = 2, n = 4: SE = s/√n = 2/2 = 1. First calculation: t = (12 - 10)/1 = 2.000000. df = n - 1 = 3. Two-sided p = 2[1 - F_t,3(2)] = 0.139326. Since 0.139326 > 0.05, fail to reject H0."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The formula permits s=0, producing a division-by-zero t statistic and undefined p-value unless the calculator explicitly rejects the input. The unspecified tail/sign convention can also change the decision.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json