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Common methods to calculate confidence band for binomial distribution

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The textbook confidence interval for binomial trial is given by

\(CI=\hat{p}\pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{\tilde{n}}}\)

It is widely recognizedthat the actual coverage probability of the standard interval is poor for p near 0 or 1.Below are some alternatives that give a much better confidence bands

**Agrestii-Coull Interval**

The confidence band is given by

\(CI_{AC}=\tilde{p}\pm z\sqrt{\frac{\tilde{p}(1-\tilde{p})}{\tilde{n}}}\)

where

\(\tilde{n}=n+z^2\)

and

\(\tilde{p}=\frac{1}{\tilde{n}}(x+\frac{z^2}{2})\)

**Wilson Interval**

\(CI_W =\frac{x+z^2/2}{n+z^2}\pm \frac{z\sqrt{n}}{n+z^2}\sqrt{\hat{p}(1-\hat{p})+z^2/(4n)}\)

**Exact Confidence Band**

For small sample size, exact confidence band should be used.

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