모분산에 대한 구간추정
ex. X_i \sim N(\mu, \sigma^2). 모수 \mu 는 알려져 있음.
\sigma^2의 100(1-\alpha)%% 신뢰구간(confidence interval)
\begin{align*}
&\hat\sigma^2 = {1 \over n}~
\sum_{i=1}^{n}~(x_i-\mu)^2
\kern{10pt} \longrightarrow \kern{10pt}
\mu\text{가 알려진 상황}
\\[20pt]
&\sum_{i=1}^{n}~
\bigg({X_i-\mu\over \sigma}\bigg)^2
=
{n\hat\sigma^2\over\sigma}
\sim \chi^2(n)
\end{align*}
\begin{align*}
1 - \alpha &= P(a<~\sigma^2~<b) =
P\bigg(~\dfrac{na}{\sigma^2}
< \dfrac{n\hat\sigma^2}{\sigma^2}
< \dfrac{nb}{\sigma^2}
\bigg)
\\[15pt]
&= P\bigg(
\chi^2~_{1-{\alpha \over 2}}(n) <
\dfrac{n\hat\sigma^2}{\sigma^2}
< \chi^2~_{\alpha \over 2}(n)
\bigg)
\end{align*}