Two Proportions

A two-proportion test is simply a check of whether two groups have different rates of something, like the share of visitors who sign up on page A versus page B. The formula has two parts. First, $\hat p$ (the pooled rate: all the sign-ups from both groups over all the visitors). Then $z$, with $\hat p_1 - \hat p_2$ (the gap between the two rates) on top and a standard error built from the pooled rate on the bottom. It's the one-proportion test from before, stretched to cover two groups.

Hypotheses: $H_0: p_1 = p_2$. Under $H_0$ the groups share one proportion, best estimated by pooling them.

Why pool? The test works out how surprising your gap would be if $H_0$ were true. If it is, both groups have the same rate, and the best estimate of that one rate uses all the data at once. That's the same logic as using $p_0$ in the one-proportion test: the standard error comes from the world $H_0$ describes. The $\frac{1}{n_1} + \frac{1}{n_2}$ part is variances adding again, one term for each group.

Conditions: at least 10 successes and 10 failures in each group, and independent random samples.

A common mistake. It's tempting to watch an A/B test as data come in and stop the moment $z$ crosses 1.96. That inflates false alarms: random wobbles cross the line now and then, and stopping at the first crossing catches them. Decide the sample size in advance, then test once.

A/B tests of sign-up rates, vaccine trials comparing infection rates, and polls comparing support between groups.