Question

7. You have one observation Y , which has one of the discrete pdf’s y f0(y) f1(y) 0 0.1 0.3 1 0.1 0.1 2 0.1 0.1 3 0.1 0.2 4 0.2 0.1 5 0.1 0.1 6 0.3 0.1 You want to test H0 : f0 is true Ha : f1 is true (a) Here is a test: reject H0 if Y = 0, 1, 2, 3, or 5. What are the probabilities of the two types of error for this test? What is the power of the test at the alternative? (b) What is the most powerful test for H0 against Ha of level α = 0.1? What is the most powerful test of level α = 0.2?

7. You have one observation Y, which has one of the discrete pdf's y fo(y) fi(y) 0.1 0.3 0.1 0.1 1 2 0.1 0.1 0.1 0.2 0.2 0.1 5 0.1 0.1 0.3 0.1 You want to test Ho fo is true Ha f is true (a) Here is a test: reject Ho if Y = 0,1,2,3, or 5. What are the probabilities of the two types of error for this test? What is the power of the test at the alternative? (b) What is the most powerful test for Ho against Ha of level a = 0.1? What is the most powerful test of level a = 0.2?

Answer 1

Here we have given a observation which has discrete pdfs.

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71 Cindete Pg. No /201 Date: For ginen one observation we want to test in z - Ho to is true Hat f. is true where Y o foly) 0.1 fily) 0.3 1 0.1 0.1 2 0.1 al 3 0.1 0.2 4 0.2 0.1 5 6 0.1 0.3. 0. l ol The test reject to it y = 0, 1, 2, 3 or 5 There are two types of error in hypothesis. testing type- I error = pl Reject Ho I Ho is true) type II erra = Placcept to I It is true) = 3 .: La Pl y = 0, 1, 2, 3 or 5 l to is true fory) = 0.17.0.1 +0.1 +0.) to .. . valuo 05 r from foly) row :. now, 32 As test is then it will rejection at accept at or 5. 01, 2, 3 y = 416 .: B= ply= 4,0 I fi is true for ")

Miracle | PG.NO: te Date: 201 B = 0.1 +0.1 - from Fily) row. B=0.2 now, to find power of test for alternative is - = Probability of correct decision = P(Rejecting Ho I Ho is false) 0.9 b) To find of level most a=0.1 powerful test for to us Ha less than It means that such test whose is or equal to da a. 16 - P(430) = 0:1 role. the test will reject Ho if y o, at Hence, this is MP test abt level a30.1. 20.2 The test je. which i Hence, level is with level o. . P x < 0.2 is at plys 1) = 0. 2 1 The test will reject to it y=0 this is most power Pul test of = 0.2 it

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