1 1 Slide © 2009 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2 2 Slide © 2009 Thomson South-Western. All Rights Reserved Chapter 14, Part A Simple Linear Regression n Simple Linear Regression Model n Least Squares Method n Coefficient of Determination

3 3 Slide © 2009 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 Slide © 2009 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 Slide © 2009 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 Slide © 2009 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 Slide © 2009 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 Slide © 2009 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 Slide © 2009 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 Slide © 2009 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 Slide © 2009 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 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 Slide © 2009 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 Slide © 2009 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 Slide © 2009 Thomson South-Western. All Rights Reserved n y -Intercept for the Estimated Regression Equation Least Squares Method

15 Slide © 2009 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 Slide © 2009 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 )  x = 10  y = 100

17 Slide © 2009 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 Slide © 2009 Thomson South-Western. All Rights Reserved n Excel Worksheet (showing data) Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation

19 Slide © 2009 Thomson South-Western. All Rights Reserved n Producing a Scatter Diagram Step 1 Select cells B1:C6 Step 2 Click the Insert tab on the Excel ribbon Step 3 In the Charts group, click Scatter Step 4 When the list of scatter diagram subtypes appears: Click Scatter with only Markers Click Scatter with only Markers Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation Step 5 In the Chart Layouts group, click Layout 1 Step 6 Select the Chart Title and replace it with Reed Auto Sales Estimated Regression Equation Auto Sales Estimated Regression Equationcontinue

20 Slide © 2009 Thomson South-Western. All Rights Reserved n Producing a Scatter Diagram Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation Step 7 Select the Horizontal Axis Title and replace it with TV Ads with TV Ads Step 8 Select the Vertical Axis Title and replace it with Cars Sold Cars Sold Step 9 Right click on the legend and click Delete Step 10 Position the mouse pointer over any Vertical Axis Major Gridline in the scatter diagram and Axis Major Gridline in the scatter diagram and right-click to display a list of options and then right-click to display a list of options and then choose Delete choose Delete

21 Slide © 2009 Thomson South-Western. All Rights Reserved n Adding the Trendline Step 13 When the Format Trendline dialog box appears: Select Trendline Options and then Select Trendline Options and then Choose Linear from the Trend/Regression Choose Linear from the Trend/Regression Type list Type list Choose Display Equation on chart Choose Display Equation on chart Click Close Click Close Step 12 Choose Add Trendline Step 11 Position the mouse pointer over any data point in the scatter diagram and right-click to display in the scatter diagram and right-click to display a list of options a list of options Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation

22 Slide © 2009 Thomson South-Western. All Rights Reserved Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation Reed Auto Sales Estimated Regression Line

23 Slide © 2009 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

24 Slide © 2009 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

25 Slide © 2009 Thomson South-Western. All Rights Reserved Coefficient of Determination r 2 = SSR/SST = 100/114 =.8772 The regression relationship is very strong; 87.72% 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.

26 Slide © 2009 Thomson South-Western. All Rights Reserved Using Excel to Compute the Coefficient of Determination n Displaying the Coefficient of Determination Step 1 Position the mouse pointer over any data point in the scatter diagram and right-click to display in the scatter diagram and right-click to display a list of options a list of options Step 2 Choose Add Trendline Step 3 When the Trendline dialog box appears: Select Trendline Options and then Select Trendline Options and then Choose Display R-squared value on chart Choose Display R-squared value on chart Click Close Click Close

27 Slide © 2009 Thomson South-Western. All Rights Reserved Using Excel to Compute the Coefficient of Determination Reed Auto Sales Estimated Regression Line

28 Slide © 2009 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

29 Slide © 2009 Thomson South-Western. All Rights Reserved The sign of b 1 in the equation is “+”. Sample Correlation Coefficient r xy =