# 409Pract2 - 1 Let X 1 X 2 X n be a random sample of size n...

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1. Let X 1 , X 2 , … , X n be a random sample of size n = 19 from the normal distribution N ( μ , σ 2 ). a) Find a rejection region of size α = 0.05 for testing H 0 : σ 2 = 30 vs. H 1 : σ 2 > 30. For which values of the sample variance s 2 should the null hypothesis be rejected? b) What is the probability of Type II Error for the rejection region in part (a) if σ 2 = 80? 2. Let X 1 , X 2 , … , X n be a random sample from N ( 0 , σ 2 ). a) Show that { ( x 1 , x 2 , … , x n ) : c x n i i = 1 2 } is the best rejection region for testing H 0 : σ 2 = 4 vs. H 1 : σ 2 = 16. b) If n = 15, find the value of c so that α = 0.05. c) If n = 15 and c is the value found in part (b), find the probability of Type II Error. 3. Let X 1 , X 2 , … , X n be a random sample from an exponential distribution with mean θ . a) Find a uniformly most powerful rejection region for testing H 0 : θ = 3 vs. H 1 : θ > 3 that is based on the statistic = n i i 1 X . That is, find a rejection region that is most powerful for testing H 0 : θ = 3 vs. H 1 : θ = θ 1 for all θ 1 > 3. b) If n = 12, use the fact that = 12 1 X 2 θ i i is χ 2 ( 24 ) to find a uniformly most powerful rejection region for testing H 0 : θ = 3 vs. H 1 : θ > 3 of size α = 0.10. c) If θ = 7, what is the power of the rejection region from part (b)?

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4. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( ) 3 2 θ 3 θ x e x x f - = x > 0 θ > 0. a) Obtain the maximum likelihood estimator of θ , θ ˆ . b) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for θ . c) Find the probability distribution of Y from part (b). d) Suppose n = 5, and x 1 = 0.2, x 2 = 1.2, x 3 = 0.2, x 4 = 0.9, x 5 = 0.3. Use part (c) to construct a 95% confidence interval for θ . e) If n = 5, find a uniformly most powerful rejection region of size α = 0.10 for testing H 0 : θ = 3 vs. H 1 : θ < 3. f) Consider the rejection region “Reject H 0 if = 5 1 3 i i x 3”. Find the significance level of this test. g) Consider the rejection region “Reject H 0 if = 5 1 3 i i x 3”. Find the power of this test at θ = 2 and θ = 1. h) Suppose n = 5, and x 1 = 0.2, x 2 = 1.2, x 3 = 0.2, x 4 = 0.9, x 5 = 0.3. Find the p-value of the test.
Answers: 1. Let X 1 , X 2 , … , X n be a random sample of size n = 19 from the normal distribution N ( μ , σ 2 ). a) Find a rejection region of size α = 0.05 for testing H 0 : σ 2 = 30 vs. H 1 : σ 2 > 30. For which values of the sample variance

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409Pract2 - 1 Let X 1 X 2 X n be a random sample of size n...

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