Question

Let X1, . . . , Xn be independent r.v.’s having the Gamma distribution with α

known and β unknown.

(i) Construct the MP test for testing the hypothesis H0: β = β1 against the

alternative HA: β = β2 (β2 > β1) at level of significance α.

(ii) By using the m.g.f. approach, show that, if X ∼ Gamma (α, β), then

X1 +°§ °§ °§+Xn ∼ Gamma (nα, β), where the Xi’s are independent and

distributed as X.

(iii) Use the CLT to carry out the test when:

n = 30, α = 10, β1 = 2.5, β2 = 3, and α = 0.05.

(iv) Compute the power of the test, also by using the CLT.