# Claim and verification record

| Claim | Scope | Evidence | Status |
|---|---|---|---|
| A binary-regressor coefficient with an intercept equals Group B mean minus Group A mean. | Exact algebra for two nonempty groups. | Derivation in `lesson.qmd`; numerical tests in `tests/statistics.test.mjs` and `tests/cross-language.test.mjs`. | Verified |
| In the same model, the HC2 coefficient variance equals `sA²/nA + sB²/nB`, the Welch variance. | Independent-group intercept-plus-binary-regressor model with at least two observations per group. | Derivation in `lesson.qmd`; leverage-adjusted residual implementations in Python, R, and Julia; JavaScript tests. | Verified |
| The pooled, Welch, and HC2 calculations agree across the included software implementations. | The checked synthetic CSV and tolerance `1e-10`. | `tests/cross-language.test.mjs` executes JavaScript, Python, R, and Julia. | Verified |
| Browser-generated groups hit the requested sample mean and standard deviation. | Exactly the exposed browser-control domain: integer `n` from 4–80, mean from 20–60, and sample standard deviation from 1–18; tolerance `1e-12`. Inputs outside that domain are rejected rather than covered by this claim. | Boundary and deterministic tests in `tests/statistics.test.mjs`. | Verified |
| The archived World Bank case has 25 LIC and 59 UMC economies, no missing 2024 values, and an unweighted UMC-minus-LIC difference of 9.6018 years with Welch 95% CI [7.4822, 11.7214]. | Archived WDI indicator SP.DYN.LE00.IN values for 2024 joined to income classifications current at 2026-08-18 retrieval; economy-level, descriptive, and unweighted. | Raw snapshot hashes, exact regeneration, coverage, and numerical tests in `tests/public-data-case.test.mjs`. | Verified |
| The default worked example identifies a causal effect. | Not claimed. It is explicitly observational. | Identification state and reporting-language tests in `tests/content-behavior.test.mjs`. | Rejected |
| Excel or Stata implementations are included. | Not claimed. | Module manifest lists only mechanically executed implementations. | Not included |

## Method references

- Student (1908), “The Probable Error of a Mean,” *Biometrika* 6(1), 1–25. [doi:10.1093/biomet/6.1.1](https://doi.org/10.1093/biomet/6.1.1)
- Welch (1947), “The Generalization of ‘Student’s’ Problem when Several Different Population Variances are Involved,” *Biometrika* 34(1/2), 28–35. [doi:10.1093/biomet/34.1-2.28](https://doi.org/10.1093/biomet/34.1-2.28)
- White (1980), “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity,” *Econometrica* 48(4), 817–838. [doi:10.2307/1912934](https://doi.org/10.2307/1912934)
- MacKinnon and White (1985), “Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties,” *Journal of Econometrics* 29(3), 305–325. [doi:10.1016/0304-4076(85)90158-7](https://doi.org/10.1016/0304-4076(85)90158-7)

These references support the methods. They do not substitute for the repository’s executable checks or establish a causal design for any application.