DSE 2012 Question 37: Point and interval estimation

Let $X_{1},X_{2},\ldots,X_{n}$ be random samples from a normal distribution with parameters $\mu$ and $\sigma^{2}$. Then the random variable $\dfrac{(n-1)S^{2}}{\sigma^{2}}=\dfrac{1}{\sigma^{2}}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}$ has a chi-squared $(\chi^{2})$ distribution with $n-1$ degrees of freedom. Here $\overline{X}$ denotes the mean of $X_{1},X_{2},\ldots,X_{n}$. Suppose area under a chi-squared curve with $n-1$ degrees of freedom to the right of $\chi^{2}_{\upsilon,n-1}$ is $\upsilon$. A $100(1-\alpha)\%$ confidence interval for the variance $\sigma^{2}$ is