Let $X_{1},..,X_{n}$ be a sample from normal distribution with mean ${\theta}$ and variance 1. Let $H_{0}:{\theta}={\theta}_{0}$ and $H_{1}:{\theta}={\theta}_{1}$. Use the Neyman-Pearson lemma to show that the best test for $H_{0}$ vs $H_{1}$ has a critical region of the form mean$(X_{1},..,X_{n}){\geq}k$.

I have done this part. The next question is:

Let ${\theta}_{0}=0$ and ${\theta}_{1}=1$. Find the values of $n$ and $k$ so that the power function of the test ${\phi}$ has values ${\phi}{\theta}_{0}=0.01$ and ${\phi}{\theta}_{1}=0.9$. Write down the corresponding decision rule.

My work thus far: Let $Y= \text{mean}(X_{1},..,X_{n})$. we have $P(Y{\geq}k|{\theta}_{0})=0.01$. Similarly for ${\theta}_{1}$. But how do I go from here?


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