Carrying Out a Two-Proportion z-Test

The null hypothesis for this test claims the two population proportions are equal: p1 = p2. If that is true, there is really just one proportion underneath both samples, so the best estimate of it combines both samples into a pooled proportion, p-hat_c = (x1 + x2) / (n1 + n2). Using this pooled value in the standard error, instead of each sample's own p-hat, makes the test assume the null hypothesis is correct while it checks how surprising the observed difference is. That assumption tightens the standard error compared to using separate proportions, which makes the test more sensitive to picking up a real difference.

The pooled standard error is SE = √(p-hat_c(1-p-hat_c)(1/n1 + 1/n2)). The z statistic is z = (p-hat1 - p-hat2) / SE, the observed gap between sample proportions divided by that pooled SE. Once you have z, the p-value comes from the standard normal curve: two-sided tests double the tail area beyond |z|, one-sided tests use just one tail, matching the direction stated in the alternative hypothesis.

This test is the natural next step after the one-proportion z-test and the two-proportion confidence interval: the interval estimates the size of a difference, while this test asks whether a difference exists at all, and the pooling step is the one piece that separates testing from estimating. It also sets up chi-square tests for comparing more than two groups, which extend this same logic of comparing observed proportions to what a shared null proportion would predict.