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Unformatted text preview: STAT 409 Homework #10 Fall 2009 (due Thursday, November 5, by 4:00 p.m.) 1 2. Bert and Ernie noticed that the following are satisfied when Cookie Monster eats cookies: (a) the number of cookies eaten during non-overlapping time intervals are independent; (b) the probability of exactly one cookie eaten in a sufficiently short interval of length h is approximately h ; (c) the probability of two or more cookies eaten in a sufficiently short interval is essentially zero. Therefore, X t , the number of cookies eaten by Cookie Monster by time t , is a Poisson process, and for any t > 0, the distribution of X t is Poisson ( t ). However, Bert and Ernie could not agree on the value of , the average number of cookies that Cookie Monster eats per minute. Bert claimed that it equals 2, but Ernie insisted that it is greater than 2. Thus, the two friends decided to test H : = 2 vs. H 1 : > 2. Bert decided to count the number of cookies Cookie Monster would eat in 5 minutes, X, and then Reject H if X is too large. Ernie, who was the less patient of the two, decided to note how much time Cookie Monster needs to eats the first 10 cookies, T, and then Reject H if T is too small. 1. a) Help Bert to find the best (uniformly most powerful) Rejection Region with the significance level of the test closest to 0.05. What is the actual value of the significance level associated with this Rejection Region? X has a Poisson ( 5 ) distribution. 0.05 = = P ( Reject H | H is true ) = P ( X c | = 2 ) = P ( Poisson ( 10 ) c ). P ( Poisson ( 10 ) 15 ) = 0.951. P ( Poisson ( 10 ) 16 ) = 0.049. Reject H if X 16 . = 0.049 . b) Find the power of the test from part (a) if = 3. Power ( = 3 ) = P ( X 16 | = 3 ) = P ( Poisson ( 15 ) 16 ) = 1 P ( Poisson ( 15 ) 15 ) = 1 0.568 = 0.432 . c) Suppose Cookie Monster ate 17 cookies in 5 minutes. Find the p-value of the test. P-value = P ( X 17 | = 2 ) = P ( Poisson ( 10 ) 17 ) = 1 P ( Poisson ( 10 ) 16 ) = 1 0.973 = 0.027 . d) Find the best randomized Rejection Region with the significance level = 0.10. P ( Poisson ( 10 ) 14 ) = 0.917. P ( Poisson ( 10 ) 15 ) = 0.083. P ( Poisson ( 10 ) = 14 ) = P ( Poisson ( 10 ) 14 ) P ( Poisson ( 10 ) 13 ) = 0.917 0.864 = 0.053. 0.083 + p 0.053 = 0.10. p 0.321. Reject H if X 15 , Reject H with probability 0.321 if X = 14 . 2. a) Help Ernie to find the best (uniformly most powerful) Rejection Region with the significance level = 0.10. Hint: If T has a Gamma ( , = 1 / ) distribution, where is an integer, then 2 T / = 2 T has a 2 ( 2 ) distribution ( a chi-square distribution with 2 degrees of freedom )....
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