A new interpretation of the rule of thumb for the Bernoulli parameter

Abstract

Interval estimation of a Bernoulli parameter p is a very fundamental and important problem in quality control area. The standard Wald confidence interval (CI) is in nearly universal use, especially in constructing control charts. Moreover, the rule of thumb “” for the standard Wald CI is introduced in most probability textbooks, despite its unsatisfactory performance with the sample size n. The constant 5 is named as the rule-of-thumb threshold for the Bernoulli parameter. This paper offers a new interpretation for the general rule of thumb “,” where the positive real value k is named as the general rule-of-thumb threshold for the Bernoulli parameter. One of our analytical results shows that “” is a necessary condition, while “” is a sufficient condition, to ensure “ and ,” respectively, where and . Moreover, this papers shows when the rule of thumb guarantees good coverage probability of the Wald interval.

Keywords:

• bernoulli parameter
• confidence interval
• coverage probability
• interval width

Acknowledgments

The author thanks the editor and the referees for their very helpful suggestions. Samping Chuang coded early versions of computer programs, which were later modified by Michael Swai-Ahn Chuang. This material is based upon work supported by the National Science Foundation under Grant NSC-102-2221-E007-087-MY2.