Let Y1,..., Yn be a random sample from the probability density function given by f (y

Chapter 10, Problem 10.98

(choose chapter or problem)

Let \(Y_{1}, \ldots, Y_{n}\) be a random sample from the probability density function given by

\(f(y \mid \theta)=\left\{\begin{array}{ll}
\left(\frac{1}{\theta}\right)^{m y^{m-1} e^{-y^{m} / \theta},}, & y>0, \\
0, & \text { elsewhere, }
\end{array}\right.\)

with m denoting a known constant.

a Find the uniformly most powerful test for testing \(H_{0}: \theta=\theta_{0}\) against \(H_{a}: \theta>\theta_{0}\).

b If the test in part (a) is to have \(\theta_{0}=100, \alpha=.05\), and \(\beta=.05\) when \(\theta_{a}=400\), find the appropriate sample size and critical region