© 2001 Prentice-Hall, Inc.Chap 13-1 BA 201 Lecture 19 Measure of Variation in the Simple Linear Regression Model (Data)Data

© 2001 Prentice-Hall, Inc. Chap 13-2 Topics Measures of Variation

© 2001 Prentice-Hall, Inc. Chap 13-3 Population Linear Regression (continued) = Random Error Y X (Observed Value of Y) = Observed Value of Y (Conditional Mean)

© 2001 Prentice-Hall, Inc. Chap 13-4 Sample Linear Regression (continued) Y X Observed Value

© 2001 Prentice-Hall, Inc. Chap 13-5 Measure of Variation: The Sum of Squares XiXi Y X Y SST =  (Y i - Y) 2 SSE =  (Y i - Y i ) 2  SSR =  (Y i - Y) 2   _ _ _

© 2001 Prentice-Hall, Inc. Chap 13-6 Measure of Variation: The Sum of Squares SST = SSR + SSE Total Sample Variability = Explained Variability + Unexplained Variability (continued)

© 2001 Prentice-Hall, Inc. Chap 13-7 Measure of Variation: The Sum of Squares SST = Total Sum of Squares Measures the variation of the Y i values around their mean Y SSR = Regression Sum of Squares Explained variation attributable to the relationship between X and Y SSE = Error Sum of Squares Variation attributable to factors other than the relationship between X and Y (continued)

© 2001 Prentice-Hall, Inc. Chap 13-8 Venn Diagrams and Explanatory Power of Regression Sales Sizes Variations in Sales explained by Sizes or variations in Sizes used in explaining variation in Sales Variations in sales explained by the error term or unexplained by Sizes Variations in store Sizes not used in explaining variation in Sales

© 2001 Prentice-Hall, Inc. Chap 13-9 The ANOVA Table in Excel ANOVA dfSSMSF Significance F RegressionpSSR MSR =SSR/p MSR/MSE P-value of the F Test Residualsn-p-1SSE MSE =SSE/(n-p-1) Totaln-1SST

© 2001 Prentice-Hall, Inc. Chap Measures of Variation The Sum of Squares: Example Excel Output for Produce Stores SSR SSE Regression (explained) df Degrees of freedom Error (residual) df Total df SST

© 2001 Prentice-Hall, Inc. Chap The Coefficient of Determination Interpretation: Measures the proportion of variation in Y that is explained by the independent variable X in the regression model (or) Measures the proportion of variation in Y that is explained by the variation in the independent variable X in the regression model

© 2001 Prentice-Hall, Inc. Chap Venn Diagrams and Explanatory Power of Regression Sales Sizes

© 2001 Prentice-Hall, Inc. Chap Coefficients of Determination (r 2 ) and Correlation (r) r 2 = 1, r 2 =.8,r 2 = 0, Y Y i =b 0 +b 1 X i X ^ Y Y i =b 0 +b 1 X i X ^ Y Y i =b 0 +b 1 X i X ^ Y Y i =b 0 +b 1 X i X ^ r = +1 r = -1 r = +0.9r = 0

© 2001 Prentice-Hall, Inc. Chap Standard Error of Estimate The standard deviation of the variation of observations around the regression line

© 2001 Prentice-Hall, Inc. Chap Measures of Variation: Produce Store Example Excel Output for Produce Stores r 2 =.94 94% of the variation in annual sales can be explained by the variability in the size of the store as measured by square footage S yx n

© 2001 Prentice-Hall, Inc. Chap Summary Described Measures of Variation