Problem 5. Suppose that we have n independent observations X1,X2, ...,Xn from N (u, 02) with 02 unknown, we want to test H0:a2=1vsH1:02>1 (1) What is the parameter space (denoted as 80) if the null hypothesis is true? What is the parameter space (denoted as 6)) if either the null hypothesis or alternative hypothesis is true? (2) Derive the maximum likelihood estimator when the null hypothesis is true. (3) Derive the maximum likelihood estimator for (u, 02) if either the null hypoth- esis or alternative hypothesis is true. m E ,0'2 (4) Compute the test statistic T = 210g (m). Simplify the test maXpLEeD EC\") statistic as a function of the sample variance 32 = 1% Zf=1(mi 517')2 (5) Form a rejection region if we use the above test statistics. (6) Assume n = 20, s2 = 2 and level a = 0.05. Compute the critical value in the rejection region. Make a decision