91Ó°ÊÓ

Find the moment-generating function for a chi square random variable and use it to show that \(E\left(\chi_{n}^{2}\right)=n\) and \(\operatorname{Var}\left(\chi_{n}^{2}\right)=2 n\).

If the random variable \(F\) has an \(F\) distribution with \(m\) and \(n\) degrees of freedom, show that \(1 / F\) has an \(F\) distribution with \(n\) and \(m\) degrees of freedom.

If a normally distributed sample of size \(n=16\) produces a \(95 \%\) confidence interval for \(\mu\) that ranges from \(44.7\) to \(49.9\), what are the values of \(\bar{y}\) and \(s\) ?