ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge 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.

A two-stage village-and-field sample

Example

Two villages are the PSUs, and the four fields in each village are its SSUs. Their fixed values are $y_{1 \cdot} = (2, 4, 6, 8), y_{2 \cdot} = (1, 3, 5, 7) .$ Select one village uniformly, then select two of its four fields uniformly without replacement. The two-stage expansion estimator has mean $36$ , the fixed total over all eight fields, and exact design variance $368 / 3$ .

Facts & Assumptions

Given: The displayed fixed field values; first-stage SRSWOR of one village from two; second-stage SRSWOR of two fields from the selected village.

[F1]

Two-stage sampling laws and the expansion estimator: The joint sample mass is the first-stage mass times the selected PSUs' second-stage masses, and $\hat{T} = \sum_{i \in S_{1}} {\hat{T}}_{i} / π_{i}^{(1)}$ with ${\hat{T}}_{i} = \sum_{j \in S_{i}} y_{i j} / π_{j ∣ i}^{(2)}$ .

[F2]

Moments, variance, and covariance on a probability space: Variance is the expectation of the centered square.

[A1]

Each stage uses the uniform SRSWOR law just stated; hence each of the six field pairs in a selected village has conditional probability $1 / 6$ and each of the twelve joint outcomes has probability $1 / 12$ .

Verification

1.1givenF1A1algebra

The village totals are $T_{1} = 20$ and $T_{2} = 16$ , $π_{1}^{(1)} = π_{2}^{(1)} = 1 / 2$ , and every second-stage field inclusion probability is $1 / 2$ ; hence a selected pair with value sum $s$ gives ${\hat{T}}_{i} = 2 s$ and $\hat{T} = 4 s$ .

2.1F1A1step 1.1algebra

Enumerating the six pairs in village 1 gives expansion estimates $(24, 32, 40, 40, 48, 56)$ and in village 2 $(16, 24, 32, 32, 40, 48)$ ; their combined sum is $432$ , so the mean of the twelve equally likely outcomes is $432 / 12 = 36 = T_{1} + T_{2}$ .

3.1F2A1step 2.1algebra∎

The twelve estimates in step 2.1 have squared deviations from $36$ summing to $1472$ ; by [F2] and their common probability $1 / 12$ , the exact design variance is $1472 / 12 = 368 / 3$ , by direct finite enumeration without an AC-dependent conditional-variance theorem.

Remarks

This finite enumeration is an independent check of one two-stage design; it does not replace the general variance proof on the A page. Holmes–Huber’s chapter supplies the experimental-design caution that repeated observations within one group do not create additional independent units; it does not prove the design-based expansion or variance formulas used here.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Shalabh, Sampling Theory, Chapter 10: Two Stage Sampling (Subsampling), PDF pp. 1–7, IIT Kanpur (standard reference, not scraped)
Sharon L. Lohr, Sampling: Design and Analysis, 2nd ed., §5.3 and §6.6, printed pp. 145–150 and 204–210 (standard reference, not scraped)
Susan Holmes and Wolfgang Huber, Modern Statistics for Modern Biology, Chapter 13, §§13.2, 13.3.2 and 13.4.5 (standard reference, not scraped)