1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 14 Simple Linear Regression n Simple Linear Regression Model n Least Squares Method n Coefficient of Determination n Model Assumptions n Testing for Significance n Using the Estimated Regression Equation for Estimation and Prediction for Estimation and Prediction

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Regression analysis can be used to develop an Regression analysis can be used to develop an equation showing how the variables are related. equation showing how the variables are related. Managerial decisions often are based on the Managerial decisions often are based on the relationship between two or more variables. relationship between two or more variables. The variables being used to predict the value of the The variables being used to predict the value of the dependent variable are called the independent dependent variable are called the independent variables and are denoted by x. variables and are denoted by x. The variable being predicted is called the dependent The variable being predicted is called the dependent variable and is denoted by y. variable and is denoted by y.

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression The relationship between the two variables is The relationship between the two variables is approximated by a straight line. approximated by a straight line. Simple linear regression involves one independent Simple linear regression involves one independent variable and one dependent variable. variable and one dependent variable. Regression analysis involving two or more Regression analysis involving two or more independent variables is called multiple regression. independent variables is called multiple regression.

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Model y =  0 +  1 x +  where:  0 and  1 are called parameters of the model,  is a random variable called the error term.  is a random variable called the error term. The simple linear regression model is: The simple linear regression model is: The equation that describes how y is related to x and The equation that describes how y is related to x and an error term is called the regression model. an error term is called the regression model.

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Equation n The simple linear regression equation is: E ( y ) is the expected value of y for a given x value. E ( y ) is the expected value of y for a given x value.  1 is the slope of the regression line.  1 is the slope of the regression line.  0 is the y intercept of the regression line.  0 is the y intercept of the regression line. Graph of the regression equation is a straight line. Graph of the regression equation is a straight line. E ( y ) =  0 +  1 x

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Equation n Positive Linear Relationship E(y)E(y)E(y)E(y) x Slope  1 is positive Regression line Intercept  0

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Equation n Negative Linear Relationship E(y)E(y)E(y)E(y) x Slope  1 is negative Regression line Intercept  0

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Equation n No Relationship E(y)E(y)E(y)E(y) x Slope  1 is 0 Regression line Intercept  0

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Estimated Simple Linear Regression Equation n The estimated simple linear regression equation is the estimated value of y for a given x value. is the estimated value of y for a given x value. b 1 is the slope of the line. b 1 is the slope of the line. b 0 is the y intercept of the line. b 0 is the y intercept of the line. The graph is called the estimated regression line. The graph is called the estimated regression line.

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved Estimation Process Regression Model y =  0 +  1 x +  Regression Equation E ( y ) =  0 +  1 x Unknown Parameters  0,  1 Sample Data: x y x 1 y 1...... x n y n b 0 and b 1 provide estimates of  0 and  1 Estimated Regression Equation Sample Statistics b 0, b 1

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Least Squares Method n Least Squares Criterion where: y i = observed value of the dependent variable for the i th observation for the i th observation^ y i = estimated value of the dependent variable for the i th observation for the i th observation

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved n Slope for the Estimated Regression Equation Least Squares Method where: x i = value of independent variable for i th observation observation_ y = mean value for dependent variable _ x = mean value for independent variable y i = value of dependent variable for i th observation observation

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved n y -Intercept for the Estimated Regression Equation Least Squares Method

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved Reed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown on the next slide. Simple Linear Regression n Example: Reed Auto Sales

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression n Example: Reed Auto Sales Number of TV Ads ( x ) TV Ads ( x ) Number of Cars Sold ( y ) 1 3 2 1 3 14 24 18 17 27  x = 10  y = 100

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved Estimated Regression Equation n Slope for the Estimated Regression Equation n y -Intercept for the Estimated Regression Equation n Estimated Regression Equation

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved Scatter Diagram and Trend Line

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved Coefficient of Determination n Relationship Among SST, SSR, SSE where: SST = total sum of squares SST = total sum of squares SSR = sum of squares due to regression SSR = sum of squares due to regression SSE = sum of squares due to error SSE = sum of squares due to error SST = SSR + SSE

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved n The coefficient of determination is: Coefficient of Determination where: SSR = sum of squares due to regression SST = total sum of squares r 2 = SSR/SST

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved Coefficient of Determination r 2 = SSR/SST = 100/114 =.8772 The regression relationship is very strong; 87.7% The regression relationship is very strong; 87.7% of the variability in the number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved Sample Correlation Coefficient where: b 1 = the slope of the estimated regression b 1 = the slope of the estimated regression equation equation

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved The sign of b 1 in the equation is “+”. Sample Correlation Coefficient r xy = +.9366

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved Assumptions About the Error Term  1. The error  is a random variable with mean of zero. 2. The variance of , denoted by  2, is the same for all values of the independent variable. all values of the independent variable. 2. The variance of , denoted by  2, is the same for all values of the independent variable. all values of the independent variable. 3. The values of  are independent. 4. The error  is a normally distributed random variable. variable. 4. The error  is a normally distributed random variable. variable.