Let \(X=\left(X_{1}, \ldots, X_{26}ight)\) and \(Y=\left(Y_{1}, \ldots, Y_{31}ight)\) represent two independent random samples from two normal population distributions. Let \(S_{X}^{2}\) and \(S_{Y}^{2}\) represent the sample variances associated with the two random samples and let \(\bar{X}\) and \(\bar{Y}\) represent the respective sample means. Define \(\hat{\sigma}^{\prime} \mathrm{s}\) as in Theorem 6.19.

(a) What is the value of \(P\left(\frac{|\bar{x}-E \bar{X}|}{\left(\hat{\sigma}_{X}^{2} / 26ight)^{1 / 2}} \leq 1.316ight)\) ?

(b) What is the value of \(P\left(\frac{26 s_{X}^{2}}{\sigma_{X}^{2}}>37.652ight)\) ?

(c) What is the value of \(P\left(s_{X}^{2}>6.02432ight)\) assuming \(\sigma_{X}^{2}=4\) ?

(d) What is the value of \(P\left(s_{Y}^{2}>1.92 s_{X}^{2}ight)\), assuming that \(\sigma_{X}^{2}=\sigma_{Y}^{2}\) ?

(e) Find the value of \(\mathrm{c}\) for which the following probability statement is true:

\(P\left(\frac{\sigma_{Y}^{2} \hat{\sigma}_{X}^{2}}{\sigma_{X}^{2} \hat{\sigma}_{Y}^{2}} \leq cight)=.05\)