Binomial Proportion Confidence Interval - Normal Approximation Interval

Normal Approximation Interval

The simplest and most commonly used formula for a binomial confidence interval relies on approximating the binomial distribution with a normal distribution. This approximation is justified by the central limit theorem. The formula is

where is the proportion of successes in a Bernoulli trial process estimated from the statistical sample, is the percentile of a standard normal distribution, is the error percentile and n is the sample size. For example, for a 95% confidence level the error is 5%, so and .

The central limit theorem applies well to a binomial distribution, even with a sample size less than 30, as long as the proportion is not too close to 0 or 1. For very extreme probabilities, though, a sample size of 30 or more may still be inadequate. The normal approximation fails totally when the sample proportion is exactly zero or exactly one. A frequently cited rule of thumb is that the normal approximation works well as long as np > 5 and n(1 − p) > 5; see however Brown et al. 2001. In practice there is little reason to use this method rather than one of the other, better performing, methods.

An important theoretical derivation of this confidence interval involves the inversion of a hypothesis test. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population proportion. The collection of values, for which the normal approximation is valid can be represented as

Since the test in the middle of the inequality is a Wald test, the normal approximation interval is sometimes called the Wald interval, but Pierre-Simon Laplace described it 1812 in Théorie analytique des probabilités (pag. 283).