Being both quality and cost conscious, a major foreign manufacturer of compact disk players is contemplating their warranty policy. The standard warranty for compact disk players sold by competing producers is 1 year. The manufacturer is considering a 2 year warranty. The operating life until failure of your compact disk player has the density

\(f(x ; \beta)=\left(x / \beta^{2}ight) \exp (-x / \beta) I_{(0, \infty)}(x)\)

for some value of \(\beta>0\), where \(x\) is measured in years.

(a) The manufacturer wants their exposure to warranty claims to be, on average, no more than 5 percent of the units sold. Find the values of \(\beta\) for which \(P(x \leq 2 ; \beta)=\int_{0}^{2} f(x ; \beta) d x \leq .05\).

(b) Based on a random sample of size 50, design a uniformly most powerful level .10 test of the null hypothesis that \(\beta\) will be in the set of values you identified in (a).

(c) A nondestructive test of the disk players that determines their operating life until failure is applied to 50 players that are randomly chosen from the assembly line. The measurements resulted in \(\bar{X}=4.27\). Test the null hypothesis in (b).

(d) Plot the power curve of this test, and interpret its meaning to the management of the manufacturing firm.