Similar to above, another choice to estimate \(\sigma ^2\) is to use a one sided confidence interval. If you want to find one of these, continue as described above but just leave one endpoint off. Indeed,
\begin{equation*}
\sigma^2 \lt E_2
\end{equation*}
can be determined using
\begin{equation*}
F(\chi^2_{\alpha} ) = F \left ( \frac{(n-1)S^2}{E_2} \right ) = \alpha
\end{equation*}
and
\begin{equation*}
E_1 \lt \sigma^2
\end{equation*}
can be determined using
\begin{equation*}
F(\chi^2_{1-\alpha} ) = F \left ( \frac{(n-1)S^2}{E_1} \right ) = 1 - \alpha.
\end{equation*}