Simple Linear Regression

1. Simple Linear Regression Chapter 14 BA 303 – Spring 2011

2. Regression • Many business decisions involve the relationship between two or more variables. • E.g., what determines sales levels? • Regression analysis is 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 used to predict the value of the dependent variable are called the independent variables and are denoted by x.

3. Simple Linear regression

4. Simple Linear Regression • Simple linear regression involves one independent variable and one dependent variable. • Approximates a straight line.

5. Simple Linear Regression Model The simple linear regression model is: y = b0 + b1x +e where: • b0and b1 are called parameters of the model, • b0 is the y-intercept, • b1 is the slope, and • eis a random variable called the error term.

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

7. E(y) x Simple Linear Regression Equation Positive Linear Relationship: Regression line Intercept b0 E(y) = 0 + 1x

8. E(y) x Simple Linear Regression Equation Negative Linear Relationship: Intercept b0 Regression line E(y) = 0- 1x

9. E(y) x Simple Linear Regression Equation No Relationship: Intercept b0 Regression line E(y) = 0 + 0x  E(y) = 0

10. Estimated Simple Linear Regression Equation The estimated simple linear regression equation • Predicted value given a value of x? • Slope? • y-intercept?

11. Estimation Process E(y) = 0 + 1x Sample Data b0 and b1 provide estimates of b0 and b1

12. Least Squares Method

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

14. _ _ x = mean value for independent variable y = mean value for dependent variable Least Squares Method 1. Slope for Estimated Regression Equation where: xi= value of independent variable for ithobservation yi = value of dependent variable for ith observation

15. Least Squares Method 2. y-Intercept for Estimated Regression Equation

16. Estimated Simple Linear Regression Equation 3. Estimated simple linear regression equation

17. Simple Linear Regression Reed Auto Sales Example

18. Scatter Plot Dependent variable (y) Independent variable (x)

19. Simple Linear Regression Reed Auto Sales e g i l f h j m a k c n d b

20. Estimated Regression Equation Slope for the Estimated Regression Equation

21. Estimated Regression Equation Slope for the Estimated Regression Equation y-Intercept for the Estimated Regression Equation

22. Estimated Regression Equation Slope for the Estimated Regression Equation y-Intercept for the Estimated Regression Equation Estimated Regression Equation

23. Applying the Estimated Regression Equation

24. Regression Line

25. Simple Linear Regression Practice

26. Practice #1 Create a scatter plot. Compute the estimated regression equation.