Theorem 10.4.3. Widest Confidence interval for p.
Given a sample of size n with relative frequency \(\tilde{p}\text{,}\) the two-sided confidence interval at confidence level \(1-\alpha\) that presumes the largest standard deviation for the unknown proportion p is given by
\begin{equation*}
\tilde{p} - z_{ \alpha/2}\frac{1}{2\sqrt{n}} \lt p \lt \tilde{p} + z_{ \alpha/2}\frac{1}{2\sqrt{n}}
\end{equation*}
where \(z_{\alpha/2}\) satisfies \(P(Z>z_{\alpha/2})=\alpha/2\) in the standard normal distribution.