Solution Found!
Let Y1, . . . , Yn be a random sample from the probability
Chapter 10, Problem 98E(choose chapter or problem)
Let be a random sample from the probability density function given by
\(f(y \mid \theta)=\left\{\begin{array}{l}
\left(\frac{1}{\theta}\right) m y^{m-1} e^{y / / \theta} y>0 \\
0,
\end{array}\right.
\)
with 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.
Equation transcription:
Text transcription:
f(y \mid \theta)=\left\{\begin{array}{l}
\left(\frac{1}{\theta}\right) m y^{m-1} e^{y / / \theta} y>0 \\
0,
\end{array}\right.
H_{0}: \theta=\theta_{0}
H_{a}: \theta>\theta_{0}
\theta_{0}=100, \alpha=.05
\beta=.05
\theta_{a}=400
Questions & Answers
QUESTION:
Let be a random sample from the probability density function given by
\(f(y \mid \theta)=\left\{\begin{array}{l}
\left(\frac{1}{\theta}\right) m y^{m-1} e^{y / / \theta} y>0 \\
0,
\end{array}\right.
\)
with 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.
Equation transcription:
Text transcription:
f(y \mid \theta)=\left\{\begin{array}{l}
\left(\frac{1}{\theta}\right) m y^{m-1} e^{y / / \theta} y>0 \\
0,
\end{array}\right.
H_{0}: \theta=\theta_{0}
H_{a}: \theta>\theta_{0}
\theta_{0}=100, \alpha=.05
\beta=.05
\theta_{a}=400
ANSWER:
Step 1 of 6
(a)
We have 
We have to find the uniformly most powerful test for
We are first going to perform the following test:-
