Statistical Inference

i) [21 marks] Assume X1, X2, ..., Xn are i.i.d. Poisson ()) random vari- ables. (Note that E(X1) = Var(Xi) = > holds then). a) [7 marks] We want to test at level a = 0.05 the hypothesis Ho : > do for n = 6, do = 1. Justify the claim that a uniformly most powerful a test for this hypothesis testing problem does exist. Also, define precisely the structure of the uniformly most powerful test. b) [7 marks] Show that this test randomly rejects Ho with a probability of 0.1787 when __Xi = 10. (Hint: You may use the fact that when ) = 1, for the random variable T = _._Xi it holds: P(T > 10) = 0.042621) c) [7 marks] Consider again Ho : A > 1 and a level a = 0.05 but the sample size n is not decided upon apriori. Using the Central Limit Theorem, find the minimal sample size for which the power of the ump 0.05 test will satisfy the requirement P(reject Hold = 2) 2 0.975