Teaching Large‐Sample Binomial Confidence Intervals

For constructing confidence intervals for a binomial proportion $p$, Simon (1996, Teaching Statistics) advocates teaching one of two large-sample alternatives to the usual $z$-intervals $\hat{p} \pm 1.96 \times S.E(\hat{p})$ where $S.E.(\hat{p}) = \sqrt{ \hat{p} \times (1 - \hat{p})/n}$. His recommendation is based on the comparison of the closeness of the achieved coverage of each system of intervals to their nominal level. This teaching note shows that a different alternative to $z$-intervals, called $q$-intervals, are strongly preferred to either method recommended by Simon. First, $q$-intervals are more easily motivated than even $z$-intervals because they require only a straightforward application of the Central Limit Theorem (without the need to estimate the variance of $\hat{p}$ and to justify that this perturbation does not affect the normal limiting distribution). Second, $q$-intervals do not involve ad-hoc continuity corrections as do the proposals in Simon. Third, $q$-intervals have substantially superior achieved coverage than either system recommended by Simon.