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Consider an independent and identically distributed random sample X1, . . . , X I}? fX HE), where the population probability mass function f); is such that: fx(mw)=:avma1f'l vm==1,2,3,.n and parameter of interest 5' > 1. The expectation is E [X] = H, the variance is Var (X) = 9(6 1), and the maximum likelihood estimator for 3 is the sample mean estimator. The expected Fisher information is n awn (a) Derive a. 95% approximate condence interval for H centred around its maxi- Ew= mum likelihood estimator. (b) Consider testing the null hypothesis 3 = 6 versus the alternative hypothesis 6' = 10. Determine the general form for an optimal rejection region, specied as a condition on the observed value of the sample mean estimator. (c) Assume that n = 60 and that the sample mean is 7.56. Using the result from (b), take a decision on whether the null hypothesis should be rejected in favour of the alternative hypothesis, at signicance level or = 0.05. (d) Calculate the numerical value of the power of the test. (e) Calculate the numerical value of the p-value for the test