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

2 2 Slide Regression n n Many business decisions involve the relationship between two or more variables. E.g., what determines sales levels? n n 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 3 Slide SIMPLE LINEAR REGRESSION

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

5 5 Slide Simple Linear Regression Model y =  0 +  1 x +  where:    0 and  1 are called parameters of the model,    0 is the y-intercept,    1 is the slope, and  is a random variable called the error term. The simple linear regression model is:

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

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

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

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

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

11 Slide Estimation Process E ( y ) =  0 +  1 x Sample Data b 0 and b 1 provide estimates of  0 and  1

12 Slide LEAST SQUARES METHOD

13 Slide Least Squares Method Least Squares Criterion where: y i = observed value of the dependent variable for the i th observation ^ y i = estimated value of the dependent variable for the i th observation

14 Slide 1. Slope for Estimated Regression Equation Least Squares Method where: x i = value of independent variable for i th observation _ y = mean value for dependent variable _ x = mean value for independent variable y i = value of dependent variable for i th observation

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

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

17 Slide Simple Linear Regression Reed Auto Sales Example TV AdsAuto Sales

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

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

Slide Estimated Regression Equation Slope for the Estimated Regression Equation

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

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

23 Slide Applying the Estimated Regression Equation x

24 Slide Regression Line

25 Slide SIMPLE LINEAR REGRESSION PRACTICE

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

27 Slide Simple Linear Regression – Practice #1

28 Slide Simple Linear Regression – Practice #1

29 Slide Simple Linear Regression – Practice #1

30 Slide Simple Linear Regression – Practice #

31 Slide