This exercise requires computation of the individual quantities to test hypotheses and set confidence intervals for (mu) and (sigma^{2}) in an assumed (mathrm{N}left(mu, sigma^{2} ight))

This exercise requires computation of the individual quantities to test hypotheses and set confidence intervals for \(\mu\) and \(\sigma^{2}\) in an assumed \(\mathrm{N}\left(\mu, \sigma^{2}\right)\) distribution.

Given the small set of data \(\{10,8,9,13,9,11\}\), do the following.

(a) Set a \(90 \%\) two-sided confidence interval for \(\mu\).

(b) Test at the \(10 \%\) level the hypothesis \(\mu=11\) versus the alternative \(\mu eq 11\).

(c) Set a \(90 \%\) lower one-sided confidence interval for \(\sigma^{2}\) (that is, a confidence interval in which the lower bound is \(-\infty\) ).

(d) Test at the \(10 \%\) level the hypothesis \(\sigma^{2} \leq 3\) versus the alternative \(\sigma^{2}>3\).