420Exam2Aans - Exam 2 1. (Version A) (White) (Answers) We...

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Exam 2 ( Version A ) ( White ) ( Answers ) 1. We wish to examine the relationship between the age of a vehicle in years ( x ) and the selling price ( y ) (in thousands of $) for a particular brand of minivan at Honest Harry’s Used Car Dealership . The data are as follows: x y 1 17 2 20 3 11 3 14 4 14 5 8 Assume that ( X , Y ) have a bivariate normal distribution. Σ x = 18, Σ y = 84, Σ x 2 = 64, Σ y 2 = 1266, Σ x y = 228, Σ ( x x ) 2 = 10, Σ ( y y ) 2 = 90, Σ ( x x ) ( y y ) = Σ ( x x ) y = – 24. a) (3) Find the sample correlation coefficient r between the age of a vehicle and the selling price. ( ) ( ) ( ) ( ) 90 10 24 2 2 - = - - - - = y y x x y y x x r 0.80 .
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1. (continued) b) (6) What is the p-value of the test H 0 : ρ = 0 vs. H a : < 0 ? (You can give a range.) Test Statistic: 2 1 2 r n r t - - = = ( ) 2 80 . 0 1 2 6 80 . 0 - - - - 2.667 . n – 2 = 4 degrees of freedom. 2.776 = t 0.025 ( 4 ) < t < t 0.05 ( 4 ) = 2.132 P-value = Area to the left of t. 0.025 < p-value < 0.05 . ( p-value 0.028 ) OR r r W - + = 1 1 2 1 ln = ( ) ( ) - - - + 80 . 0 1 80 . 0 1 2 1 ln 1.0986. Under H 0 , 0 0 ρ 1 ρ 1 2 1 ln - + = W μ = - + 0 1 0 1 2 1 ln = 0, 3 1 2 - = n W σ = 3 1 . Test Statistic: W W W z - = = 3 1 0 0986 . 1 - - 1.90 . P-value = left tail = P ( Z < – 1.90 ) = 0.0287 . c) (6) What is the p-value of the test H 0 : = – 0.5 vs. H a : < – 0.5 ? r r W - + = 1 1 2 1 ln = ( ) ( ) - - - + 80 . 0 1 80 . 0 1 2 1 ln 1.0986. Under H 0 , 0 0 ρ 1 ρ 1 2 1 ln - + = W = ( ) ( ) - - - + 50 . 0 1 50 . 0 1 2 1 ln 0.5493, 3 1 2 - = n W = 3 1 . Test Statistic: W W W z - = = 3 1 5493 . 0 0986 . 1 + - 0.95 . P-value = left tail = P ( Z < – 0.95 ) = 0.1711 .
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2. Suppose a complete second-order model Y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + β 4 x 1 2 + β 5 x 2 2 + ε was fit to n = 30 data points. > sum( lm( y ~ 1 )$residuals^2 ) [1] 600 > sum( lm( y ~ x1 + x2 )$residuals^2 ) [1] 126 > sum( lm( y ~ x1 + x2 + I(x1*x2) )$residuals^2 ) [1] 117 > sum( lm( y ~ x1 + x2 + I(x1*x2) + I(x1^2) +I(x2^2) )$residuals^2 ) [1] 96 a) (10) Perform the “significance of the regression” test at a 5% level of significance. H 0 : β 1 = β 2 = β 3 = β 4 = β 5 = 0 vs H 1 : at least one of β 1 , β 2 , β 3 , β 4 , β 5 is not zero.
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This note was uploaded on 12/17/2010 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois, Urbana Champaign.

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420Exam2Aans - Exam 2 1. (Version A) (White) (Answers) We...

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