Confidence Interval for a Difference of Proportions

When you care about whether two groups differ, the question is about the gap itself, not about each group in isolation. Building one interval around phat1 - phat2 answers that question directly: every value inside the interval is a plausible size for the true difference p1 - p2. Comparing two separate intervals instead and checking whether they overlap throws away information, because that overlap check doesn't account correctly for the combined variability of the two samples.

The point estimate is phat1 - phat2. Each sample proportion has its own variability, phat(1-phat)/n, and because the two samples are independent, their variances add. The standard error is √( phat1(1-phat1)/n1 + phat2(1-phat2)/n2 ). The interval is (phat1 - phat2) ± z·times that standard error, with z* = 1.96 for 95 percent confidence. Conditions still matter: both samples need to be random, independent of each other, and large enough that each group has at least 10 successes and 10 failures.

This builds directly on the one-proportion interval from earlier in the unit, just applied to a new parameter, the difference p1 - p2. It sets up the next step in the unit, the significance test for a difference of proportions, where instead of estimating the gap you test a specific claim about it, usually that p1 - p2 = 0. The interval and the test use the same standard error logic, just aimed at different questions: estimation versus decision.