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. Least Squares Coefficient Estimators b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared residuals (errors), SSE: Differential calculus is used to obtain the coefficient estimators b 0 and b 1 that minimize SSE Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-2 11.3

3. Least Squares Coefficient Estimators The slope coefficient estimator is And the constant or y-intercept is The regression line always goes through the mean x, y (continued) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-3

4. Computer Computation of Regression Coefficients The coefficients b 0 and b 1, and other regression results in this chapter, will be found using a computer Hand calculations are tedious Statistical routines are built into Excel Other statistical analysis software can be used Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-4

5. Interpretation of the Slope and the Intercept b 0 is the estimated average value of y when the value of x is zero (if x = 0 is in the range of observed x values) b 1 is the estimated change in the average value of y as a result of a one- unit change in x Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-5

6. Simple Linear Regression Example A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet) A random sample of 10 houses is selected Dependent variable (Y) = house price in $1000s Independent variable (X) = square feet Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-6

7. Sample Data for House Price Model House Price in $1000s (Y) Square Feet (X) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-7

8. Graphical Presentation House price model: scatter plot Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-8

9. Regression Using Excel Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-9 Excel will be used to generate the coefficients and measures of goodness of fit for regression Data / Data Analysis / Regression

10. Regression Using Excel Data / Data Analysis / Regression Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-10 (continued) Provide desired input:

11. Excel Output 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 The regression equation is: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall (continued) Ch. 11-12

13. Graphical Presentation House price model: scatter plot and regression line Slope = 0.10977 Intercept = 98.248 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-13

14. Interpretation of the Intercept, b 0 b 0 is the estimated average value of Y when the value of X is zero (if X = 0 is in the range of observed X values) Here, no houses had 0 square feet, so b 0 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-14

15. Interpretation of the Slope Coefficient, b 1 b 1 measures the estimated change in the average value of Y as a result of a one- unit change in X Here, b 1 =.10977 tells us that the average value of a house increases by.10977($1000) = $109.77, on average, for each additional one square foot of size Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-15