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The cost of using exact confidence intervals for a binomial proportion. (English) Zbl 1348.62092
Summary: When computing a confidence interval for a binomial proportion $$p$$ one must choose between using an exact interval, which has a coverage probability of at least $$1-\alpha$$ for all values of $$p$$, and a shorter approximate interval, which may have lower coverage for some $$p$$ but that on average has coverage equal to $$1-\alpha$$. We investigate the cost of using the exact one and two-sided Clopper-Pearson confidence intervals rather than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper-Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper-Pearson interval and Bayesian intervals based on noninformative priors.

##### MSC:
 62F25 Parametric tolerance and confidence regions 62F12 Asymptotic properties of parametric estimators
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