Exact Confidence Limits for Binomial Proportions—Pearson and Hartley Revisited

It has long been known that there is a problem with the results of ‘conventional’ (as epitomized in the Biometrika Tables for Statisticians) techniques for calculating confidence limits for parameters of discrete distributions. Specifically, the calculated limits at a given confidence level are always much too wide, i.e. overly conservative. For large sample sizes (n≥ 100) this is not important, but for small samples, the conventional techniques can be very conservative. In this note, exact confidence limits for the parameter p, as calculated in binomial sampling, are presented. A Bayesian technique is used, and the results are presented for the situation where no prior information is assumed, corresponding to the ‘conventional’ scenario for confidence‐limit estimation. Our results are compared quantitatively with those in the Biometrika Tables by use of Monte‐Carlo simulation. The results show, as expected, that for small sample sizes, the Biometrika Tables yield confidence intervals that are considerably too wide, and that our approach does indeed produce exact confidence limits. Extensive graphs of results are presented.