Regression analysis can be used to develop an

1. Chapter 12 - Simple Linear Regression • Managerial decisions often are based on the relationship between two or more variables. • Regression analysis can be used to develop an equation showing how the variables are related. • The variable being predicted is called the dependent variable and is denoted by y. • The variables being used to predict the value of the dependent variable are called the independent variables and are denoted by x. • Simple linear regression involves one independent • variable and one dependent variable. Two or more • independent variables is called multiple regression. • The relationship between the two variables is approximated by a straight line.

2. Simple Linear Regression Model • The equation that describes how y is related to x and • an error term is called the regression model. • The simple linear regression model is: y = b0 + b1x +e where: • b0 and b1 are called parameters of the model, • e is a random variable called the error term.

3. Simple Linear Regression Equation • The simple linear regression equation is: E(y) = 0 + 1x • Graph of the regression equation is a straight line. • b0 is the y intercept of the regression line. • b1 is the slope of the regression line. • E(y) is the expected value of y for a given x value.

4. E(y) x Simple Linear Regression Equation • Positive Linear Relationship Regression line Intercept b0 Slope b1 is positive

5. E(y) x Simple Linear Regression Equation • Negative Linear Relationship Intercept b0 Regression line Slope b1 is negative • You can show No Relationship

6. is the estimated value of y for a given x value. Estimated Simple Linear Regression Equation • The estimated simple linear regression equation • The graph is called the estimated regression line. • b0 is the y intercept of the line. • b1 is the slope of the line.

7. Sample Data: x y x1 y1 . . . . xnyn Estimated Regression Equation Sample Statistics b0, b1 Estimation Process Regression Model y = b0 + b1x +e Regression Equation E(y) = b0 + b1x Unknown Parameters b0, b1 b0 and b1 provide estimates of b0 and b1

8. ^ yi = estimated value of the dependent variable for the ith observation Least Squares Method where: yi = observed value of the dependent variable for the ith observation • Least Squares Criterion

9. Simple Linear Regression Model y Observed Value of y for xi εi Slope = β1 Predicted Value of y for xi Random Error for this xi value Intercept = β0 x Xi

10. _ x = mean value for independent variable y = mean value for dependent variable _ Least Squares Method where: xi = value of independent variable for ith observation yi = value of dependent variable for ith observation • y-Intercept for the Estimated Regression Equation • Slope for the Estimated Regression Equation

11. Simple Linear Regression • Example: Reed Auto Sales 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 here. Number of TV Ads (x) Number of Cars Sold (y) 1 3 2 1 3 -1x-6 1x4 0x-2 -1x-3 1x7 14 24 18 17 27 Sx = 10 Sy = 100

12. Estimated Regression Equation • Slope for the Estimated Regression Equation • y-Intercept for the Estimated Regression Equation • Estimated Regression Equation

13. Reed Auto Sales Estimated Regression Line Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation

14. Measures of Variation • Total variation is made up of two parts: Total Sum of Squares Regression Sum of Squares Error Sum of Squares where: = Mean value of the dependent variable yi = Observed value of the dependent variable = Predicted value of y for the given xi value

15. Measures of Variation • SST = total sum of squares (Total Variation) • Measures the variation of the yi values around their mean y • SSR = regression sum of squares (Explained Variation) • Variation attributable to the relationship between x and y • SSE = error sum of squares (Unexplained Variation) • Variation in y attributable to factors other than x

16. Measures of Variation y yi  y  y _ _ y y x xi

17. Coefficient of Determination r2 or R2 SST = SSR + SSE where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error r2 = SSR/SST = 100/114 = .8772 The regression relationship is very strong; 87.72% 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. • Relationship Among SST, SSR, SSE

18. where: b1 = the slope of the estimated regression equation Sample Correlation Coefficient – We learned in Chapter 3

19. The sign of b1 in the equation is “+”. Sample Correlation Coefficient rxy = +.9366 Note: This only holds for simple regression

20. Examples of Approximate r2 (or R2) Values Y r2 = 1 Perfect linear relationship between X and Y: 100% of the variation in Y is explained by variation in X X r2 = 1 Y X r2 = 1

21. Examples of Approximate r2 (or R2) Values Y 0 < r2 < 1 Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X X Y X

22. Examples of Approximate r2 (or R2) Values r2 = 0 Y 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) X r2 = 0

23. Armand’s Pizza