A Note on Teaching 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.