Confidence Intervals for a Proportion

Definition Suppose the number of successes $X$ in $n$ trials follows $X \sim \operatorname{Binomial}(n, p)$, which requires a fixed number of independent trials that share one success probability $p$, and let $\hat{p} = X/n$. Write $z = z_{1 - \alpha/2}$ for the standard normal quantile with upper tail area $\alpha/2$, so that $z = z_{0.975} \approx 1.96$ for 95 percent confidence. The Wald interval for $p$ is

It comes from the normal approximation to the distribution of $\hat{p}$, with the observed $\hat{p}$ substituted for the unknown $p$ in the standard error $\sqrt{p(1 - p)/n}$. Every interval on this page relies on the binomial model, so the model has to match how the data were collected. Outcomes repeated on the same person, a success probability that drifts, or a selected sample break it, and no interval formula repairs that. For a random sample drawn without replacement from a finite population, the binomial model is an approximation that works well when the sample is a small fraction of the population.

Definition The nominal coverage of an interval method is the confidence level $1 - \alpha$ it is designed for, such as 95 percent. Its actual coverage at a value $p$ is the probability, computed under the model with that $p$ as the true value, that the interval produced from the data contains $p$. For a method that turns each count $x$ into an interval $[L(x), U(x)]$, the actual coverage is $C(p) = P_p(L(X) \le p \le U(X))$, which equals the sum of binomial probabilities

where $S(p)$ is the set of counts whose interval contains $p$. Coverage describes the procedure before the data are seen. Once an interval has been computed it either contains $p$ or it does not, and the 95 percent is not a probability that this particular interval is correct.

Definition Under the same binomial model, the Wilson interval, also called the score interval, is the set of all $p$ in $[0, 1]$ that pass the normal comparison

in which the standard error is computed from the candidate $p$ rather than from $\hat{p}$. Squaring turns this into a quadratic inequality in $p$, and its solution is the interval $[c - h, c + h]$ with

The endpoints always lie in $[0, 1]$, and when $0 < X < n$ the interval contains $\hat{p}$ in its interior. The method still rests on a normal approximation, so its coverage is approximate.

Definition Under the binomial model, observe $X = x$ and write $P_p$ for binomial probabilities computed with success probability $p$. For $0 < x < n$, the equal-tailed Clopper-Pearson interval at level $1 - \alpha$ is $[p_L, p_U]$, where the endpoints solve

When $x = 0$ the lower endpoint is $0$ and the upper endpoint solves $(1 - p_U)^n = \alpha/2$. When $x = n$ the upper endpoint is $1$ and the lower endpoint solves $p_L^n = \alpha/2$. Each tail probability changes continuously and strictly with $p$, so each equation has exactly one solution. The endpoints can also be computed as quantiles of beta distributions. Under the binomial model the actual coverage of this procedure is at least $1 - \alpha$ for every $p$ in $[0, 1]$. It is often called the exact binomial interval, because it uses binomial probabilities instead of a normal approximation.

Definition A boundary proportion is an observed $\hat{p}$ equal to $0$ or $1$, which happens when $X = 0$ or $X = n$. With $X = 0$ and level $1 - \alpha$, the Wald interval is the single point $0$, the Wilson interval is $[0, z^2/(n + z^2)]$, and the Clopper-Pearson interval is $[0, 1 - (\alpha/2)^{1/n}]$. With $X = n$ the intervals are the mirror images, reflected about $\tfrac12$. Under the model every $p < 1$ gives $X = 0$ positive probability, namely $(1 - p)^n$, so a zero count does not show that $p = 0$.