The pressure of air (in \(\mathrm{MPa}\) ) entering a compressor is measured to be \(X=8.5 \pm 0.2\), and the pressure of the air leaving the compressor is measured to be \(Y=21.2 \pm 0.3\). The intermediate pressure is therefore measured to be \(P=\sqrt{X Y}=13.42\). Assume that \(X\) and \(Y\) come from normal populations and are unbiased.

a. From what distributions is it appropriate to simulate values \(X^{*}\) and \(Y^{*}\) ?

b. Generate simulated samples of values \(X^{*}, Y^{*}\), and \(P^{*}\).

c. Use the simulated sample to estimate the standard deviation of \(P\).

d. Construct a normal probability plot for the values \(P^{*}\). Is it reasonable to assume that \(P\) is approximately normally distributed?

e. If appropriate, use the normal curve to find a \(95 \%\) confidence interval for the intermediate pressure.