Maximal Interval:
\begin{equation*}
P( 0.36 - 1.960 \frac{1}{2\sqrt{400}} \lt p \lt 0.36 + 1.960 \frac{1}{2\sqrt{400}} ) = 1 - \alpha.
\end{equation*}
or
\begin{equation*}
P( 0.36 - 1.960 \frac{1}{40} \lt p \lt 0.36 + 1.960 \frac{1}{40} ) = 1 - \alpha.
\end{equation*}
or
\begin{equation*}
P( 0.311 \lt p \lt 0.409 ) = 1 - \alpha.
\end{equation*}
Notice the interval is only slightly wider than when using \(\tilde{p}\) to estimate p in the first case.