Theorem 10.4.7. Determining Sample Size for proportions with a preliminary estimate.
Given a margin of error E and preliminary relative frequency estimate \(\tilde{p_0}\) the sample size needed to create the corresponding confidence interval is given by
\begin{equation*}
n \gt \left ( \frac{z_{\alpha /2}}{E} \right )^2 \tilde{p_0}(1-\tilde{p_0}).
\end{equation*}