Two-Sample Problems

In Part 1, we estimated a single population mean—the proportion of home wins, voter turnout, average income. Now we compare means between groups. Does turnout differ between Black and non-Black voters? Do people who receive a threatening letter vote more often than those who don’t? Do college graduates earn more than non-graduates?

These are two-sample problems: we have two subpopulations and want to estimate the difference in their means. The estimator is simple—a difference of subsample means—but the analysis requires new tools. We need to think about conditional expectations and variances, and we need to be careful about what’s random and what’s fixed when we condition on group membership.