Question: Let \({{X}_{1}},...,{{X}_{n}}\) be i.i.d observations from a \(N\left( \mu ,{{\sigma }^{2}} \right)\) distribution with known variance \({{\sigma }^{2}}\).

1. Formulate the likelihood ratio test for testing Ho: µ = 2 vs H1 : µ > 2 (10 points)
2. Write down the equations needed to determine the sample size n and the critical value c such that the size of the test is 0.05 and the probability of type-II error when \(\mu =2.5\) is 0.1.
3. If we took 16 samples and observed \(\bar{X}\) =2.5, and the population standard deviation is 0.5, compute the p-value based on the data for the test in a). Will you reject H0 at level \(\alpha \) = 0.05 based on the p-value calculated? (4 points)

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