## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 8, Number 1 (2014), 817-840.

### The cost of using exact confidence intervals for a binomial proportion

#### Abstract

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.

#### Article information

**Source**

Electron. J. Statist., Volume 8, Number 1 (2014), 817-840.

**Dates**

First available in Project Euclid: 16 June 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1402927499

**Digital Object Identifier**

doi:10.1214/14-EJS909

**Mathematical Reviews number (MathSciNet)**

MR3217790

**Zentralblatt MATH identifier**

1348.62092

**Subjects**

Primary: 62F25: Tolerance and confidence regions

Secondary: 62F12: Asymptotic properties of estimators

**Keywords**

Asymptotic expansion binomial distribution confidence interval expected length sample size determination proportion

#### Citation

Thulin, Måns. The cost of using exact confidence intervals for a binomial proportion. Electron. J. Statist. 8 (2014), no. 1, 817--840. doi:10.1214/14-EJS909. https://projecteuclid.org/euclid.ejs/1402927499