# Finding a Uniformily Most Powerful Region

1. Mar 5, 2010

### cse63146

1. The problem statement, all variables and given/known data

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 $$H_0:\theta=75 \ VS \ H_1:\theta>75$$

2. Relevant equations

3. 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?

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