420Hw06ans - STAT 420 Spring 2010 Homework#6(12 points(due...

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STAT 420 Spring 2010 Homework #6 (12 points) (due Friday, March 12, by 4:00 p.m.) 1. Do NOT use a computer for this problem. Homework #5 Problem 5 (continued) An employee claims that drinking beer has no effect on the amount of time it takes for him to perform a particular task. The following data show how many seconds he took to perform the task after consuming various quantities of beer, measured in ounces: Beer Consumption, x Time, y 0 62 12 50 24 59 36 74 48 59 60 83 72 68 Consider the model Y i = α + β x i + ε i ., where ε i ’s are i.i.d. N ( 0, σ 2 ) . Recall: Σ x = 252, x = 36, Σ y = 455, y = 65, Σ x 2 = 13,104, Σ y 2 = 30,295, Σ x y = 17,388, SXX = Σ ( x x ) 2 = 4032, SYY = Σ ( y y ) 2 = 720, SXY = Σ ( x x ) ( y y ) = 1,008. y ˆ = 56 + 0.25 x . SSRes = ( 2 ˆ i i y y - = 468. SSReg = ( - 2 ˆ y y i = 252. 2 e s = 93.6. R 2 = 0.35.

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h) Construct a 95% confidence interval for β . Confidence interval for β : SXX 2 α t β ˆ e s ± α = 0.05 . α / 2 = 0.025 . n – 2 = 5 degrees of freedom. t 0.025 = 2.571 . 4032 6747 . 9 571 . 2 25 . 0 ± 0.25 ± 0.39 ( 0.14 , 0.64 ) i) Test the employee’s claim at a 10% level of significance. That is, test H 0 : β = 0 vs. H 1 : β > 0 at a 10% level of significance. T = 4032 6747 . 9 0 25 . 0 SXX 0 β β ˆ - = - e s = 1.641 . n – 2 = 5 degrees of freedom. Rejection Region: T > t α . α = 0.10 t 0.10 = 1.476 . The Test Statistic T IS in the Rejection Region. Reject H 0 at α = 0.10. t distribution critical values Upper-tail probability P-value = right tail. df 0.10 0.05 0.05 < p-value < 0.10 . 5 1.476 2.015 ( p-value 0.0809 ) j) Test H 0 : β = 0 vs. H 1 : β 0 at a 10% level of significance. T = 4032 6747 . 9 0 25 . 0 SXX 0 β β ˆ - = - e s = 1.641 . n – 2 = 5 degrees of freedom. Rejection Region: T < – t α / 2 . or T > t α / 2 . α = 0.10 t 0.05 = 2.015 . The Test Statistic T is NOT in the Rejection Region. Accept H 0 at α = 0.10. P-value = 2 tails. 0.10 < p-value < 0.20 . ( p-value 0.1618 )
k) The supervisor believes that when the employee is not drinking beer, it takes him on average 60 seconds to perform the task. Test H 0 : α = 60 vs. H 1 : α 60 at a 10% level of significance. T = ( 29 ( 29 4032 36 0 7 1 6747 . 9 60 56 SXX 0 1 2 2 0 α α ˆ x n s e - + - = - + - = 0.606778 . n – 2 = 5 degrees of freedom.

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