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Unformatted text preview: STAT 420 Spring 2010 Homework #7 (12 points) (due Friday, March 19, by 4:00 p.m.) 1. A students wonders if people of similar heights tend to date each other. She measures herself, her dormitory roommate, and the women in the adjoining rooms; then she measures the next man each woman dates. Here are the data (heights in inches): Women, x 68 64 68 67 70 65 Men, y 73 68 71 69 73 66 x = 402, y = 420, x 2 = 26,958, y 2 = 29,440, x y = 28,167, ( x x ) 2 = 24, ( y y ) 2 = 40, ( x x ) ( y y ) = ( x x ) y = 27. Assume that ( X , Y ) have a bivariate normal distribution. a) Find the sample correlation coefficient r between the heights of the women and men. ( 29 ( 29 ( 29 ( 29 40 24 27 2 2 = = y y x x y y x x r 0.87142 . b) Test H : = 0 vs. H 1 : 0. Use = 0.05. What is the pvalue of this test ( you can give a range ) ? Test Statistic: T = 2 2 87142 . 1 2 6 87142 . 1 2 = r n r 3.553 . Rejection Region: Reject H if T < t 0.025 or T > t 0.025 n 2 = 4 degrees of freedom t 0.025 ( 4 ) = 2.776 . Reject H : = 0 at = 0.05. t 0.025 ( 4 ) = 2.776 < 3.553 < 3.747 = t 0.01 ( 4 ) pvalue = 2 tails. 0.02 < pvalue < 0.05 ( pvalue 0.02556 ) c) What is the pvalue of the test H : = 0.3 vs. H 1 : > 0.3 ? r r + = 1 1 2 1 W ln =  + 87142 . 1 87142 . 1 2 1 ln = 1.33895. Under H , W 1 1 2 1 ln + = =  + 30 . 1 30 . 1 2 1 ln = 0.30952, 3 1 2 W  = n = 3 1 . Test Statistic: W W W Z = = 3 1 30952 . 33895 . 1 = 1.783 . Pvalue = right tail = P ( Z > 1.783 ) = 0.0375 . d) What is the pvalue of the test H : = 0.5 vs. H 1 : 0.5 ? r r + = 1 1 2 1 W ln =  + 87142 . 1 87142 . 1 2 1 ln = 1.33895. Under H , W 1 1 2 1 ln + = =  + 50 . 1 50 . 1 2 1 ln = 0.54931, 3 1 2 W  = n = 3 1 . Test Statistic: W W W Z = = 3 1 54931 . 33895 . 1 = 1.368 . Pvalue = two tails = 2 P ( Z > 1.368 ) = 2 0.0853 = 0.1706 . e) Construct a 95% confidence interval for . 100 ( 1 ) % confidence interval for : + + 1 1 , 1 1 b b a a e e e e , where 3 2 1 1 2 ln + = n r r a z , 3 2 1 1 2 ln + + = n r r b z . 3 2 1 1 2 ln + = n r r a z = 2.6779 3 96 . 1 2 = 0.4147....
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This note was uploaded on 04/08/2012 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Stepanov

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