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

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Stat 312: Lecture 21 Correlations Moo K. Chung [email protected] April 10, 2003 Concepts 1. The coefficient of determination r 2 = S 2 xy /S xx S yy . 2. The sample correlation coefficient r is defined as r = S xy / p S xx S yy . In Lecture 20, we showed that 0 r 2 1. Hence - 1 r 1. 3. r = 0 if and only if paired observations ( x j , y j ) pass through the same straight line. From Lec- ture 20 example 2, if ( x j , y j ) pass through a cir- cle, r = 0. So r can be used to measure if the paired observations show linear relationship. 4. Population correlation coefficient ρ ( X, Y ) = Cov ( X, Y ) p Var ( X ) Var ( Y ) . In-class problems Example 1. Is there any linear relationship between the mathematical achievement test scores ( x i ) and calculus grades ( y i ) for 10 college freshmen? x k 39 , 43 , 21 , 64 , 57 , 47 ,
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Unformatted text preview: , 34 , 52 y k 65 , 78 , 52 , 82 , 92 , 89 , 73 , 98 , 56 , 75 > cor(x,y) [1] 0.8397859 Example 2. Is there any relationship between the number oF times you miss submitting homeworks ( x i ) and the fnal grade ( y i )? x k 03 , 04 , 01 , 02 , 01 , 00 , 00 , 07 , 08 , 05 y k 70 , 75 , 80 , 75 , 85 , 95 , 90 , 70 , 50 , 60 > cor(x,y) [1] -0.8938321 20 30 40 50 60 70 60 70 80 90 x y ±igure 1: Complete lack oF linear ft. 2 4 6 8 50 x ±igure 2: Complete lack oF linear ft. Self-study problems Please read P.529. Example 12.14. Homework 7 Due April 29. 11:00am. 12.30., 12.36., 12.58., 12.60.,12.66....
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