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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.

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.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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.

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