Lecture23 - lecture 23, Linear Regression IV 1 / 22 The...

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Unformatted text preview: lecture 23, Linear Regression IV 1 / 22 The Sample Covariance The Correlation Coefficient Relationship Between the Correlation Coefficient and the Simple Coefficient of Determina- tion A Review Example lecture 23, Linear Regression IV Outline 1 The Sample Covariance 2 The Correlation Coefficient 3 Relationship Between the Correlation Coefficient and the Simple Coefficient of Determination 4 A Review Example lecture 23, Linear Regression IV 2 / 22 The Sample Covariance The Correlation Coefficient Relationship Between the Correlation Coefficient and the Simple Coefficient of Determina- tion A Review Example lecture 23, Linear Regression IV The Sample Covariance The Sample Covariance • Suppose two variables x and y are observed in pairs, and the observations are ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) • The sample covariance between x and y is cov ( x , y ) = ∑ n i = 1 ( x i- ¯ x )( y i- ¯ y ) n- 1 = SXY n- 1 • cov ( x , y ) > 0: x and y tend to move in the (same/opposite) direction(s) • cov ( x , y ) < 0: x and y tend to move in the (same/opposite) direction(s) lecture 23, Linear Regression IV 3 / 22 The Sample Covariance The Correlation Coefficient Relationship Between the Correlation Coefficient and the Simple Coefficient of Determina- tion A Review Example lecture 23, Linear Regression IV The Sample Covariance Example Q: Is cov ( x , y ) positive or negative? lecture 23, Linear Regression IV 4 / 22 The Sample Covariance The Correlation Coefficient Relationship Between the Correlation Coefficient and the Simple Coefficient of Determina- tion A Review Example lecture 23, Linear Regression IV The Correlation Coefficient The Correlation Coefficient • Suppose two variables x and y are observed in pairs, and the observations are ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) • The correlation coefficient (or just correlation) between x and y is r = cov ( x , y ) s x s y , where cov ( x , y ) = SXY / ( n- 1 ) is the sample covariance between x and y ; s x = p SXX / ( n- 1 ) and s y = p SYY / ( n- 1 ) are the sample standard deviations of x and y respectively. • Equivalently, r = SXY √ SXX · SYY lecture 23, Linear Regression IV 5 / 22 The Sample Covariance The Correlation Coefficient Relationship Between the Correlation Coefficient and the Simple Coefficient of Determina- tion A Review Example lecture 23, Linear Regression IV The Correlation Coefficient Scatter Plot of Fuel Consumption Data ● ● ● ● ● ● ● ● 20 30 40 50 60 70 6 8 10 12 14 TEMP FUEL lecture 23, Linear Regression IV 6 / 22 The Sample Covariance The Correlation Coefficient Relationship Between the Correlation Coefficient and the Simple Coefficient of Determina- tion A Review Example lecture 23, Linear Regression IV The Correlation Coefficient Fuel Consumption Case Temp 28.0 28.0 32.5 39.0 45.9 57.8 58.1 62.5 Fuel Cons 12.4 11.7 12.4 10.8 9.4 9.5 8.0 7.5 • SXY =- 179 . 6475 , SXX = 1404 . 355 , and SYY = 25 . 549 • Hence the correlation coefficient r is r = SXY √ SXX · SYY =- 179 . 6475 √ 1404 . 355 × 25 . 549 =- . 948 . lecture 23,...
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This note was uploaded on 12/28/2010 for the course BUSINESS A ISOM 111 taught by Professor Yingyingli during the Fall '10 term at HKUST.

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Lecture23 - lecture 23, Linear Regression IV 1 / 22 The...

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