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We consider confidence interval estimation for a binomial proportion when the data have already been observed and x, the observed number of successes in a sample of size n, is zero. In this case, the main objective of the investigator is usually to obtain a reasonable upper bound for the true probability of success, i.e., the upper limit of a one-sided confidence interval. In this article, we use observed interval length and p-confidence to evaluate eight methods for finding the upper limit of a confidence interval for a binomial proportion when x is known to be zero. Long-run properties such as expected interval length and coverage probability are not applicable because the sample data have already been observed. We show that many popular approximate methods that are known to have good long-run properties in the general setting perform poorly when x=0 and recommend that the Clopper-Pearson exact method be used instead.

On Finding the Upper Confidence Limit for a Binomial Proportion when Zero Successes are Observed