4 min readStatistical Inference
The 95% Interval That Covers 81% of the Time
The textbook confidence interval for a proportion has exact coverage you can compute by summing over the n+1 possible samples, and at n = 30 with p = 0.10 it is 0.8085 rather than 0.95. Coverage does not improve monotonically with n, and in a rare-event setting it can fall to 0.0392. Two one-line alternatives fix it.
Statistics