Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 22 Inference on Correlation Moo K. Chung [email protected] April 17, 2003 Concepts 1. Assume that X and Y are two random variables whose joint probability density function f is the bivariate normal such that X N ( μ X , σ 2 X ) , Y N ( μ Y , σ 2 Y ) and ρ ( X, Y ) = ρ . In this situation, we can show that ( Y | X = x ) = β 0 + β 1 x , where β 1 = ρσ Y X . This is the regression model we studied! 2. Testing H 0 : ρ = 0 vs. H 1 : ρ 6 = 0 is equiva- lent to H 0 : β 1 = 0 vs. H 1 : β 1 6 = 0 . So the appropriate test statistic for testing H 0 : ρ = 0 is T = ˆ β 1 /S ˆ β 1 t n - 2 (see Lecture 19, Concept 3). This test statistic is equivalent to T = r n - 2 1 - r 2 . 3. Testing for independence is equivalent to testing ρ = 0 in bivariate data. In-class problems Exercise 12.67. A sample of 10,000 ( x i , y i ) pairs re- sulted in the sample correlation coefficient r = 0 . 022 . Test H 0 : ρ = 0 vs. H 1 : ρ 6 = 0 at level 0.05. Also est if x i and y i are independent. Solution. > r<-0.022 > r*sqrt(10000-2)/sqrt(1-rˆ2) [1] 2.200313 > pt(2.2,9998) [1] 0.9860852 Using Concept 2, t = 2 . 2 . P -value = P ( | T | > 2 . 2) = 2(1 - 0 . 986) = 0 . 028 . So we reject H 0 . Since independence is equivalent to zero correlation in bivariate data, we can use the same test
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