Question

Let X1, X2, . . . , X12 denote a random sample of size 12 from...

Let X1, X2, . . . , X12 denote a random sample of size 12 from Poisson distribution with mean θ.
a) Use Neyman-Pearson Lemma to show that the critical region defined by
(12∑i=1) Xi, ≤2
is a best critical region for testing H0 :θ=1/2 against H1 :θ=1/3.
b.) If K(θ) is the power function of this test, find K(1/2) and K(1/3). What is the significance level, the probability of the 1st type error, the probability of the 2nd type error
for this test?
c) Show that the critical region C defined by
(12∑i=1) Xi, ≤ 2
is a uniformly most powerful one for testing H0 :θ=1/2 against H1 :θ<1/2.

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