Setting Up a Two-Proportion Test

A test comparing two proportions always starts from a null hypothesis that says there's no difference: H0: p1 = p2. That assumption is the whole point. If you assume the two groups truly share one common proportion, you can estimate that shared proportion by combining both samples into one, treating them as if they came from the same population. This combined estimate is the pooled proportion, p-hat_c = (x1 + x2) / (n1 + n2), the total successes over the total sample size.

The pooled proportion gets used two places: checking that np and n(1-p) are at least 10 for each group (using the pooled value in place of each sample's own proportion), and computing the standard error for the test statistic. Both steps lean on the null's assumption that one shared p actually exists. The alternative hypothesis, Ha, says p1 does not equal p2 (or states a direction, p1 > p2 or p1 < p2), and conditions also require independent random samples, from independent groups, with large enough counts.

A confidence interval sits right next to this test but works differently. An interval never assumes the two proportions are equal — it's built to estimate the actual size of p1 - p2, equal or not. So it keeps each sample's own p-hat1 and p-hat2 separately in its standard error instead of pooling them. Seeing why pooling disappears there is the cleanest way to remember why it appeared here in the first place: pooling is a consequence of assuming equality, nothing more.