Finding a Uniformily Most Powerful Region

[cse63146]

Homework Statement

Let $$X_1,...,X_{25}$$ be from a sample of 25 from a normal distribution $$N(\theta,100)$$. Find a UMP region with $$\alpha = 0.1$$ for testing $$H_0:\theta=75 \ VS \ H_1:\theta>75$$

Homework Equations

The Attempt at a Solution

So after performing the Likelihood ratio test, I determined the critial region to be $$(\Sigma X_i => K)$$ where K is some constant.

Using the CLT $$\frac{\Sigma X_i - n\mu}{\sigma \sqrt{n}}$$

$$P(\Sigma X_i => K) =P_{H_0}( \frac{\Sigma X_i - 25(75)}{10\sqrt{75}} => \frac{K - 25(75)}{10\sqrt{75}} )=0.1$$

$$\frac{K - 1875}{86.6}=1.28$$ solving for K = 1985.848

Have I made a mistake anywhere?

 

[nate808]

Is this the correct UMP region?



Your approach to finding the UMP region for this hypothesis testing problem is correct. You have correctly used the likelihood ratio test and the central limit theorem to determine the critical region. Your calculation for the constant K also appears to be correct. Therefore, your answer for the UMP region is also correct. Good job!