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Suppose the random variable, X, follows an exponential distribution with mean 0 (0 ≤ 0 <∞). Let X₁, X2,..., X₁ be a random sample of size n from the population of X. (a) Find the rejection region for a most powerful test of Ho: 0= 0o versus Ha: 0= 0a < 0o. (b) Is the above test uniformly most powerful? Explain your answer with argument. (c) Show (or argue) that the distribution of 1X₁ = ²nX is xn (which can be used to find the constant associated with the rejection region of the above test). (d) Let = 2 and n = 1. Determine the value of k (that is used in the specification of the rejection region) so that a = 0.05. (e) If Y₁, Y2,..., Ym is a random sample from another exponential distribution with mean 6, find the likelihood ratio criterion for testing Ho: 0= 6 versus H₁ : 08. (Note here you need to use both samples - X₁, X2,..., X₁ and Y₁, Y2,..., Ym to test the equality of the two population means.) (f) Using the sampling distributions of 3nX and 3mY under Ho: 0= 6, show that the above test in part (e) is based on an F statistic. [Hint: Both statistics follow x² distributions independently.]

Suppose the random variable, X, follows an exponential distribution with mean 0 (0 ≤ 0 <∞). Let X₁, X2,..., X₁ be a random sample of size n from the population of X. (a) Find the rejection region for a most powerful test of Ho: 0= 0o versus Ha: 0= 0a < 0o. (b) Is the above test uniformly most powerful? Explain your answer with argument. (c) Show (or argue) that the distribution of 1X₁ = ²nX is xn (which can be used to find the constant associated with the rejection region of the above test). (d) Let = 2 and n = 1. Determine the value of k (that is used in the specification of the rejection region) so that a = 0.05. (e) If Y₁, Y2,..., Ym is a random sample from another exponential distribution with mean 6, find the likelihood ratio criterion for testing Ho: 0= 6 versus H₁ : 08. (Note here you need to use both samples - X₁, X2,..., X₁ and Y₁, Y2,..., Ym to test the equality of the two population means.) (f) Using the sampling distributions of 3nX and 3mY under Ho: 0= 6, show that the above test in part (e) is based on an F statistic. [Hint: Both statistics follow x² distributions independently.]

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 3CR
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Suppose the random variable, X, follows an exponential distribution with mean 0 (0 ≤ <∞0). Let X₁, X2,..., X₁ be a random sample of size n from the population of X. (a) Find the rejection region for a most powerful test of Ho: 0 = 0o versus H₁ : 0 = 0 < 00. (b) Is the above test uniformly most powerful? Explain your answer with argument. (c) Show (or argue) that the distribution of Σ₁=1 X; = ²nX is X²n (which can be used to find the constant associated with the rejection region of the above test). (d) Let 80 = 2 and n = 1. Determine the value of k (that is used in the specification of the rejection region) so that a = 0.05. (e) If Y₁, Y2,..., Ym is a random sample from another exponential distribution with mean 6, find the likelihood ratio criterion for testing Ho: 0= 6 versus H₁ : 08. (Note here you need to use both samples - X₁, X2,..., X and Y₁, Y₂,..., Ym to test the equality of the two population means.) (f) Using the sampling distributions of nX and mỸ under H₁ : 0 = 6, show that the above test in part (e) is based on an F statistic. [Hint: Both statistics follow x² distributions independently.]
Transcribed Image Text:Suppose the random variable, X, follows an exponential distribution with mean 0 (0 ≤ <∞0). Let X₁, X2,..., X₁ be a random sample of size n from the population of X. (a) Find the rejection region for a most powerful test of Ho: 0 = 0o versus H₁ : 0 = 0 < 00. (b) Is the above test uniformly most powerful? Explain your answer with argument. (c) Show (or argue) that the distribution of Σ₁=1 X; = ²nX is X²n (which can be used to find the constant associated with the rejection region of the above test). (d) Let 80 = 2 and n = 1. Determine the value of k (that is used in the specification of the rejection region) so that a = 0.05. (e) If Y₁, Y2,..., Ym is a random sample from another exponential distribution with mean 6, find the likelihood ratio criterion for testing Ho: 0= 6 versus H₁ : 08. (Note here you need to use both samples - X₁, X2,..., X and Y₁, Y₂,..., Ym to test the equality of the two population means.) (f) Using the sampling distributions of nX and mỸ under H₁ : 0 = 6, show that the above test in part (e) is based on an F statistic. [Hint: Both statistics follow x² distributions independently.]
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