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420Hw05ans - STAT 420(due Friday October 3 by 3:00 p.m...

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STAT 420 Fall 2008 Homework #5 (due Friday, October 3, by 3:00 p.m.) 1. 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 Σ x = 252, Σ y = 455, Σ x 2 = 13,104, Σ y 2 = 30,295, Σ x y = 17,388, Σ ( x x ) 2 = 4032, Σ ( y y ) 2 = 720, Σ ( x x ) ( y y ) = Σ ( x x ) y = 1,008. Consider the model Y i = α + β x i + ε i ., where ε i ’s are i.i.d. N ( 0, σ 2 ) . a) Find the equation of the least-squares regression line. 7 252 = = ° n x x = 36 . 7 455 = = ° n y y = 65 . 4032 1008 SXX SXY ° ˆ = = = 0.25 . x y - = ° ˆ ± ˆ = 65 – 0.25 36 = 56 . Least-squares regression line: y ˆ = 56 + 0.25 x .
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b) Give an estimate for σ , the standard deviation of the observations about the true regression line? x y y ˆ e e 2 0 62 56 6 36 12 50 59 9 81 24 59 62 3 9 36 74 65 9 81 48 59 68 9 81 60 83 71 12 144 72 68 74 6 36 Sum: 0 468 SSE OR In class we showed that since ( ) ( ) ± ˆ ˆ ° ˆ ° ˆ ° ˆ ° ˆ x x y x y x y i i i i x - + = + - = + = , RSS = ( ) ( ) ( ) 2 2 2 2 2 2 ° ˆ ° ˆ ° ˆ ˆ = - = - = - ° ° ° x x x x y y i i i SXX = 252 .
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