Santner, Thomas J.
(1997):
A Note on Teaching Binomial Confidence Intervals.
Collaborative Research Center 386, Discussion Paper 87

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Abstract
For constructing confidence intervals for a binomial proportion $p$, Simon (1996, Teaching Statistics) advocates teaching one of two largesample 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 adhoc continuity corrections as do the proposals in Simon. Third, $q$intervals have substantially superior achieved coverage than either system recommended by Simon.