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# 410Hw11ans - STAT 410 Homework#11(due Wednesday August 5 by...

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P ( Poisson ( 10 ) 16 ) = 0.049 . Reject H 0 if X 16 . α = 0.049 . b) Find the power of the test from part (a) if λ = 3. Power ( λ = 3 ) = P ( X 16 | λ = 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 | λ = 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 α = 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 0 if X 15 , Reject H 0 with probability 0.321 if X = 14 . 2. a) Help Ernie to find the best (uniformly most powerful) Rejection Region with the significance level α = 0.10. Hint: If T has a Gamma ( α , θ = 1 / λ ) distribution, where α is an integer, then 2 T / θ = 2 λ T has a χ 2 ( 2 α ) distribution ( a chi-square distribution with 2 α degrees of freedom ) . T has a Gamma distribution with parameters α = 10 and θ = 1 / λ .
2 λ T has a chi-square distribution with 2 α = 20 degrees of freedom 0.10 = α = P ( Reject H 0 | H 0 is true ) = P ( T c | λ = 2 ) = P ( 4 T 4 c | λ = 2 ) = P ( χ 2 ( 20 ) 4 c ) . 4 c = ( 29 20 2 90 . 0 χ = 12.44. c = 3.11. Reject H 0 if T 3.11 . b) Ernie decided to Reject H 0 if it takes Cookie Monster less than 2.5 minutes to eat the first 10 cookies. Find the significance level α for this Rejection Region.

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410Hw11ans - STAT 410 Homework#11(due Wednesday August 5 by...

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