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day7 - So far we have developed graphical and numerical...

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So far we have developed graphical and numerical methods for describing a single variable. Measuring Association between Two Variables Example : The following data show the median expenditures (in millions of dollars) and shipments in millions of barrels for 10 major brands of beer. Brand Media Expenditures Shipments Budweiser 120 36.3 Bud Light 68.7 20.7 Miller Lite 100.1 15.9 Coors Light 76.6 13.2 Busch 8.7 8.1 Natural Light 0.1 7.1 Miller Genuine Draft 21.5 5.6 Miller High Lite 1.4 4.4 Busch Light 5.3 4.3 Milwaukee’s Best 1.7 4.3 We can make a scatter diagram to graphically summarize the relationship between media expenditures and shipments of barrels. 1

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We would also like a numerical measure of the relationship. Covariance If we have n observations from a sample, ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ), the sample covariance is s xy = ( x i - ¯ x )( y i - ¯ y ) n - 1 Ex : What is the sample covariance for the beer data? x y x i - ¯ x y i - ¯ y ( x i - ¯ x )( y i - ¯ y ) 120 36.3 68.7 20.7 100.1 15.9 76.6 13.2 8.7 8.1 0.1 7.1 21.5 5.6 1.4 4.4 5.3 4.3 1.7 4.3 2
Interpretation of Covariance 3

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If the covariance is negative, there is a negative linear relationship.
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day7 - So far we have developed graphical and numerical...

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