Let \(X\) be a random sample of size \(\mathrm{n}\) from a \(N\left(\mu, \sigma^{2}ight)\) population distribution representing the yield per acre, in pounds, of a new strain of hops used in the production of premium beer.

(a) Justify that the random interval \(\left(n S^{2} / \chi_{\alpha}^{2}, n S^{2} / \chi_{1-\alpha}^{2}ight)\) will have an outcome that contains the value of the population variance \(\sigma^{2}\) with probability \(\alpha\), where \(\chi_{\delta}^{2}\) is a number for which \(P\left(x>\chi_{\delta}^{2}ight)=\delta\) when \(X\) has a \(\chi^{2}\) distribution with \(n-1\) degrees of freedom.

(b) Suppose that \(s^{2}=9\) and \(n=20\). Define a confidence interval that is designed to have a .95 probability of generating an outcome that contains the value of the population variance of hop yields, \(\sigma^{2}\). Generate a confidence interval outcome for the variance.