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.

Proportional and disproportionate stratified allocations

Example

Let $U_{1}$ have fixed values $7, 9, 11, 13$ and let $U_{2}$ have fixed values $18, 22$ . Compare independent stratified SRSWOR allocations $(n_{1}, n_{2}) = (2, 1)$ and $(1, 2)$ , both with total sample size three. The first allocation is proportional to stratum sizes; the second oversamples the smaller stratum.

Facts & Assumptions

Given: The six fixed values above, independent within-stratum SRSWOR, and the two stated allocations.

[F1]

Stratified sampling designs and the stratified total estimator: The estimator is ${\hat{T}}_{s t} = N_{1} {\overset{ˉ}{y}}_{1, R} + N_{2} {\overset{ˉ}{y}}_{2, R}$ ; certainty strata have zero variance contribution.

[F2]

Unbiasedness and variance of the stratified total estimator: The estimator is design-unbiased, and its variance is the sum of the stratum contributions $N_{h}^{2} (1 - n_{h} / N_{h}) S_{h}^{2} / n_{h}$ over noncertainty strata.

Verification

1.1givenchoosealgebra

The fixed means are ${\overset{ˉ}{y}}_{1} = 10$ and ${\overset{ˉ}{y}}_{2} = 20$ . The population variances are $S_{1}^{2} = ((- 3)^{2} + (- 1)^{2} + 1^{2} + 3^{2}) / 3 = 20 / 3$ and $S_{2}^{2} = ((- 2)^{2} + 2^{2}) / 1 = 8$ , so the total is $4 (10) + 2 (20) = 80$ .

2.1F1F2step 1.1

Both allocations satisfy $1 \leq n_{h} \leq N_{h}$ . By [F2] and step 1.1, each corresponding weighted estimator has expectation $80$ .

2.2F2step 1.1algebra

Under allocation $(2, 1)$ , [F2] and step 1.1 give variance $16 (1 - 2 / 4) (20 / 3) / 2 + 4 (1 - 1 / 2) 8 / 1 = 80 / 3 + 16 = 128 / 3$ .

2.3F1F2step 1.1algebra

Under allocation $(1, 2)$ , the second stratum is a census, so [F1], [F2], and step 1.1 give variance $16 (1 - 1 / 4) (20 / 3) / 1 = 80$ .

3.1step 2.1step 2.2step 2.3algebra∎

Step 2.1 gives unbiasedness for both designs, and steps 2.2–2.3 give $128 / 3 < 80$ for these fixed values.

Depends on

Used by

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Sources

Shalabh, Sampling Theory, MTH 417 lecture notes, Chapter 4: Stratified Sampling, PDF pp. 8–9 and 12–14 (standard reference, not scraped)
Robin Arnold, STAT392 Sample Surveys, Chapter 8: Stratified Sampling, §§8.3 and 8.8 (standard reference, not scraped)