420Exam1Aans(2) - Exam 1 1. (Version A) (White) (Answers) x...

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Exam 1 ( Version A ) ( White ) ( Answers ) 1. A substance used in biological and medical research is shipped by air freight to users in cartons of 1,000 ampules. The data, involving eight shipments, were collected on the number of times the carton was transferred from one aircraft to another (including the initial loading) over the shipment route ( x ) and the number of ampules found to be broken upon arrival ( y ). Assume the simple linear regression model is appropriate: Y i = α + β x i + ε i ., where ε i ’s are i.i.d. N ( 0, σ 2 ). x y 2 16 1 9 3 17 1 12 4 22 2 13 1 8 2 15 Σ x = 16, Σ y = 112, Σ x 2 = 40, Σ y 2 = 1712, Σ x y = 256, Σ ( x x ) 2 = 8, Σ ( y y ) 2 = 144, Σ ( x x ) ( y y ) = Σ ( x x ) y = 32. a) (5) Find the equation of the least-squares regression line. 8 16 = = n x x = 2. 8 112 = = n y y = 14. 8 32 SXX SXY β ˆ = = = 4 . x y - = β ˆ α ˆ = 14 – 4 2 = 6 . Least-squares regression line: y ˆ = 6 + 4 x . b) (3) Give an estimate for σ 2 . SSRegr = 2 β ˆ SXX = 4 2 8 = 128. SSResid = SYY – SSRegr = 144 – 128 = 16. s e 2 = 2 SSResid - n = 6 16 2.6667 . OR 2 ˆ σ = n SSResid = 8 16 = 2 .
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c) (5) Find the p-value of the test H 0 : β = 3 vs. H a : β > 3. (You may give a range.) Test Statistic: SXX 0 β β ˆ e s t - = = 8 6 16 3 4 - 1.732 . n – 2 = 6 df. 1.440 = t 0.10 ( 6 ) < t < t 0.05 ( 6 ) = 1.943 P-value = Area to the right of t . 0.05 < p-value < 0.10 . ( p-value 0.067. ) d) (5) Find the p-value of the test H 0 : α = 3 vs. H a : α 3. (You may give a range.) Test Statistic: ( ) ( ) 8 2 8 1 6 16 3 6 SXX 1 2 2 0 α α ˆ x n s t e + - + - = = 2.3238
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420Exam1Aans(2) - Exam 1 1. (Version A) (White) (Answers) x...

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