Why the textbook interval for a proportion misbehaves

Question

You observe 50 conversions out of 1,000 visitors, a 5% rate, and want a 95% confidence interval for the true conversion rate. The standard "Wald" interval is $\hat p \pm 1.96\sqrt{\hat p(1-\hat p)/n}$ .

Compute the Wald interval, then explain when it breaks down and what to use instead.

Accepted Answer

The Wald interval. With \hat p = 0.05 and n = 1000: 	ext{SE} = \sqrt{\frac{0.05 	imes 0.95}{1000}} = \sqrt{0.00004750} \approx 0.00689, 	ext{CI} = 0.05 \pm 1.96 	imes 0.00689 = 0.05 \pm 0.0135 = [0.0365,\ 0.0635]. Here n is large and \hat p isn't near 0 or 1, so this interval is fine. Where Wald breaks. The Wald interval centers on \hat p and uses \hat p itself to estimate the standard error. Two failure modes: 1. Small n or extreme \hat p. With few events (say 2 conversions out of 40), the normal approximation is poor and the true coverage of a "95%" Wald interval can drop well below 95%…

Why the textbook interval for a proportion misbehaves

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