ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An exact unknown-variance t interval for a four-point sample

Example

Assume the Axiom of Choice and an iid sample of size $4$ from $N (μ, σ^{2})$ with $σ > 0$ . For the observed values $(1, 2, 4, 5)$ , the sample mean is $\overset{ˉ}{x} = 3$ and the unbiased sample variance is $s^{2} = 10 / 3$ . The exact $95 %$ t interval produced by the sample is

[3 - t_{3, 39 / 40} \frac{\sqrt{10 / 3}}{2}, 3 + t_{3, 39 / 40} \frac{\sqrt{10 / 3}}{2}] .

The repeated-sampling coverage of this interval procedure is exactly $0.95$ under the stated normal model; this is a coverage statement about the procedure, not a probability assigned to the fixed parameter after observing these data.

Facts & Assumptions

Given: The stated normal sample model and observed values $(1, 2, 4, 5)$ .

[F1]

For an iid normal sample with unknown variance and sample size $n \geq 2$ , the t pivot has law $t_{n - 1}$ and its central quantile interval has exact coverage $1 - α$ . Normal sample mean and residual are independent, giving the exact t pivot

[F2]

The $u$ -quantile $t_{k, u}$ is the unique point where the continuous strictly increasing $t_{k}$ CDF equals $u$ . Student t and Fisher F CDFs are continuous and strictly increasing on their supports

Verification

technique · direct

1.1givenalgebra

The arithmetic mean is $(1 + 2 + 4 + 5) / 4 = 3$ . The deviations from $3$ are $- 2, - 1, 1, 2$ , whose squared sum is $4 + 1 + 1 + 4 = 10$ .

2.1F1F2step 1.1algebra∎

Dividing this sum by $n - 1 = 3$ gives $s^{2} = 10 / 3$ , hence $s = \sqrt{10 / 3}$ . Substituting $n = 4$ , $α = 5 / 100$ , and the observed sample into [F1] yields the displayed interval with the unique quantile $t_{3, 39 / 40}$ from [F2]; its repeated-sampling coverage is $1 - α = 95 / 100$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Dmitry Panchenko, MIT 18.443 Statistics for Applications, Fall 2003, Lecture 17 (standard reference, not scraped)
Joshua M. Tebbs, STAT 512 Lecture Notes, Chapter 8, §8.7.1 (standard reference, not scraped)