Reading an Interval That Contains Zero

A confidence interval for a difference in proportions, p1 - p2, is a range of plausible values for how far apart the two true proportions really are. Zero means no difference at all. If zero falls inside the interval, 'no difference' is one of the plausible values, so you don't have evidence that the groups differ. If zero falls outside the interval, every plausible value is positive or every plausible value is negative, which tells you which group is higher, and by roughly how much.

The position of zero relative to the interval is what decides the story. If the whole interval sits above zero, p1 is plausibly larger than p2 for every value in the range. If the whole interval sits below zero, p2 is plausibly larger. If zero sits inside, somewhere between the low and high end, the data are consistent with p1 being bigger, p2 being bigger, or them being equal, so you can't pick a direction with confidence.

This connects directly to a two-proportion z-test for p1 = p2. A confidence interval containing zero at a given confidence level lines up with failing to reject the null hypothesis at the matching significance level: both say the observed difference could plausibly be due to sampling variation alone. An interval missing zero lines up with rejecting the null: both say the difference looks real. Reading the interval is often faster than running the test, since the interval also tells you the direction and a plausible size of the effect, not just whether it's significant.