A Note on Teaching Large–Sample Binomial Confidence Intervals

This paper addresses the question of which large–sample binomial confidence interval method to teach in elementary courses. Recently, Goodall (1995), Simon (1996), and the references therein, have discussed the merits of several large–sample systems of binomial intervals. Three of the systems considered these articles are the z-interval, a “t-based interval,” and a continuitycorrected interval, the latter denoted by c-interval herein. The 95% nominal z limits are p̂± 1.96× S.E.(p̂) (0.1) where S.E.(p̂) = √ p̂× (1− p̂)/n is an estimate of the standard error of p̂. The t-intervals widen the (0.1) limits by using a larger critical point; they are p̂ ± t × S.E.(p̂), where t is the two-sided upper-0.05 critical point of the t-distribution with n− 1 degrees of freedom. The c-intervals are p̂± 1.96× {S.E.(p̂) + 1/2n} ; (0.2) they also widen z-intervals but do so by increasing S.E.(p̂) with a “continuity correction.” Simon (1996) compares the achieved coverage of these systems for sample sizes 5 to 40. He concludes “the t-based interval achieves better coverage than the z-based interval,” and furthermore that the c-intervals are an attractive alternative to t-intervals. This article suggests that a third alternative to z-intervals, called q-intervals herein, should be strongly preferred in elementary courses to either tor c-intervals. First, q-intervals are more easily motivated than z-intervals because they require only a straightforward application of the Central Limit Theorem (without the need to estimate the variance of p̂ and to justify that this perturbation does not affect the normal limiting distribution). Second, the q-intervals do not involve ad-hoc continuity corrections. Third, q-intervals have substantially superior achieved coverage than either the t or the c systems. For teachers who are interested in learning more advanced material on this subject, there are several systems of “exact” intervals that have been proposed for the binomial problem. Here exact