Statistics for Business and Economics 8 th Edition Chapter 11 Simple Regression Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-1

Explanatory Power of a Linear Regression Equation Total variation is made up of two parts: Total Sum of Squares Regression Sum of Squares Error (residual) Sum of Squares where: = Average value of the dependent variable y i = Observed values of the dependent variable i = Predicted value of y for the given x i value Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch

Analysis of Variance 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 linear relationship between x and y SSE = error sum of squares Variation attributable to factors other than the linear relationship between x and y Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-3

Analysis of Variance (continued) xixi y X yiyi SST =  (y i - y) 2 SSE =  (y i - y i ) 2  SSR =  (y i - y) 2  _ _ _ y  Y y _ y  Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch Explained variation Unexplained variation

Coefficient of Determination, R 2 The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable The coefficient of determination is also called R-squared and is denoted as R 2 note: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-5

Examples of Approximate r 2 Values r 2 = 1 Y X Y X Perfect linear relationship between X and Y: 100% of the variation in Y is explained by variation in X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-6

Examples of Approximate r 2 Values Y X Y X 0 < r 2 < 1 Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-7

Examples of Approximate r 2 Values r 2 = 0 No linear relationship between X and Y: The value of Y does not depend on X. (None of the variation in Y is explained by variation in X) Y X r 2 = 0 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-8

Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations10 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet % of the variation in house prices is explained by variation in square feet Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-9

Correlation and R 2 The coefficient of determination, R 2, for a simple regression is equal to the simple correlation squared Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch

Estimation of Model Error Variance An estimator for the variance of the population model error is Division by n – 2 instead of n – 1 is because the simple regression model uses two estimated parameters, b 0 and b 1, instead of one is called the standard error of the estimate Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch

Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations10 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch

Comparing Standard Errors YY X X s e is a measure of the variation of observed y values from the regression line The magnitude of s e should always be judged relative to the size of the y values in the sample data i.e., s e = $41.33K is moderately small relative to house prices in the $200 - $300K range Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch