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

2. 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. 11-2 11.4

3. 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

4. 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. 11-4 Explained variation Unexplained variation

5. 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

6. 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

7. 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

8. 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

9. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 58.08% 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

10. 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. 11-10

11. 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. 11-11

12. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-12

13. 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. 11-13