Justifying a Claim About Two Proportions

A gap between two sample proportions, like 52% versus 47%, can happen just from the luck of who got sampled, even if the true proportions are equal. That's why you don't trust the point estimate alone. You build a confidence interval for p1 - p2, which gives a whole range of plausible values for the true difference, and you let that range do the talking.

To justify a claim from the interval, state what the interval is (values of p1 - p2 you're confident contains the true difference), then check whether zero is inside it. If zero is in the interval, zero is a plausible value for the true difference, so you cannot conclude the proportions differ. If the entire interval is above zero or entirely below zero, every plausible value says one group's true proportion is bigger, so you have convincing evidence of a real difference. Always connect this back to the context: which group, and in which direction.

This sits right next to significance testing for two proportions: the interval version and the test version agree, since an interval missing zero lines up with a small p-value. It also sets up chi-square tests, where you compare more than two groups or more than two categories at once instead of just checking one interval against zero.