Wilson Score Interval: Let’s do this on in parts...
\begin{equation*}
z_{\alpha/2} \sqrt{\frac{\tilde{p}(1-\tilde{p}) + \frac{z_{\alpha/2}^2}{4n}}{n}} = 1.96 \sqrt{ \frac{0.36 \cdot 0.64 + \frac{1.96^2}{1600}}{400}} \approx 0.04728
\end{equation*}
Therefore,
\begin{equation*}
\frac{0.36 + \frac{1.96^2}{800} - 0.04728}{1 + \frac{1.96^2}{400}} \lt p \lt \frac{0.36 + \frac{1.96^2}{800} + 0.04728}{1 + \frac{1.96^2}{400}}
\end{equation*}
or
\begin{equation*}
0.3145 \lt p \lt 0.4082
\end{equation*}
which is slightly different than the first and slightly smaller than the second.