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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $X$ have the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and unknown $p \\in [0, 1]$, and let $\\alpha \\in (0, 1)$.\nA *binomial proportion confidence interval* of level $1 - \\alpha$ is an interval $[L(X), U(X)]$ with endpoints computed from $X$ alone.\nIts coverage function is $C(p) = P_p\\big(L(X) \\le p \\le U(X)\\big)$.\nIt is a [[confidence interval|0K4NMMJBG2]] of level at least $1 - \\alpha$ when $C(p) \\ge 1 - \\alpha$ for every $p \\in [0, 1]$.\nIt is an approximate interval when $C(p)$ tends to $1 - \\alpha$ as $n \\to \\infty$, for each fixed $p \\in (0, 1)$.\n\n## Remarks\n\nLet $\\hat p = X/n$ and let $z$ be the $1 - \\alpha/2$ quantile of the standard normal law.\nThe Wald interval $\\hat p \\pm z \\sqrt{\\hat p(1 - \\hat p)/n}$ is an approximate interval, by the normal approximation to the binomial law.\nThe Wilson interval is the set of $p$ with $\\abs{\\hat p - p} \\le z \\sqrt{p(1 - p)/n}$; it is also approximate, with better coverage for small $n$.\nThe Clopper–Pearson interval has level at least $1 - \\alpha$ for every $p$; its endpoints are quantiles of beta laws.\nNon-example: the Wald interval is not of level at least $1 - \\alpha$ for all $p$.\nIf $X = 0$, it is the single point $\\set{0}$, so for small $p \u003e 0$ the coverage $C(p)$ is close to $P_p(X \\ge 1) = 1 - (1 - p)^n$, which tends to $0$ as $p \\to 0$.\n",
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