In the full sample, covariate adjustment moves the estimated coefficient from $0.73$ to $0.32$, a large change consistent with substantial measured <confounding>. In the Catholic-middle-school subsample the estimates remain between $0.48$ and $0.60$, supporting the claim that restriction has already improved comparability. The cost is visible in the standard errors, which rise from about $0.08$--$0.09$ to $0.13$--$0.15$ despite similar coefficient magnitudes.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 4 / b / i / Solution

 Solution (source code)