1 1 Slide © 2004 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University.

2 2 Slide © 2004 Thomson/South-Western Chapter 12 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 n Computer Solution n Residual Analysis: Validating Model Assumptions

3 3 Slide © 2004 Thomson/South-Western 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.

4 4 Slide © 2004 Thomson/South-Western 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

5 5 Slide © 2004 Thomson/South-Western Simple Linear Regression Equation n Positive Linear Relationship E(y)E(y)E(y)E(y) E(y)E(y)E(y)E(y) xx Slope  1 is positive Regression line Intercept  0

6 6 Slide © 2004 Thomson/South-Western Simple Linear Regression Equation n Negative Linear Relationship E(y)E(y)E(y)E(y) E(y)E(y)E(y)E(y) xx Slope  1 is negative Regression line Intercept  0

8 8 Slide © 2004 Thomson/South-Western 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.

9 9 Slide © 2004 Thomson/South-Western 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

10 Slide © 2004 Thomson/South-Western 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

12 Slide © 2004 Thomson/South-Western n y -Intercept for the Estimated Regression Equation Least Squares Method where: x i = value of independent variable for i th observation observation n = total number of observations _ y = mean value for dependent variable _ x = mean value for independent variable y i = value of dependent variable for i th observation observation

13 Slide © 2004 Thomson/South-Western Example: Reed Auto Sales n Simple Linear Regression 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.

17 Slide © 2004 Thomson/South-Western n Producing a Scatter Diagram Step 1 Select cells B1:C6 Step 2 Select the Chart Wizard Step 3 When the Chart Type dialog box appears: Choose XY (Scatter) in the Chart type list Choose XY (Scatter) in the Chart type list Choose Scatter from the Chart sub-type display Choose Scatter from the Chart sub-type display Click Next > Click Next > Step 4 When the Chart Source Data dialog box appears Click Next > Click Next > Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation

18 Slide © 2004 Thomson/South-Western n Producing a Scatter Diagram Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation Step 5 When the Chart Options dialog box appears: Select the Titles tab and then Select the Titles tab and then Delete Cars Sold in the Chart title box Enter TV Ads in the Value (X) axis box Enter Cars Sold in the Value (Y) axis box Select the Legend tab and then Select the Legend tab and then Remove the check in the Show Legend box Click Next >

19 Slide © 2004 Thomson/South-Western n Producing a Scatter Diagram Step 6 When the Chart Location dialog box appears: Specify the location for the new chart Specify the location for the new chart Select Finish to display the scatter diagram Select Finish to display the scatter diagram Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation

20 Slide © 2004 Thomson/South-Western n Adding the Trendline Step 3 When the Add Trendline dialog box appears: On the Type tab select Linear On the Type tab select Linear On the Options tab select the Display On the Options tab select the Display equation on chart box Click OK Click OK Step 2 Choose the Add Trendline option Step 1 Position the mouse pointer over any data point and right click to display the Chart point and right click to display the Chart menu menu Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation

22 Slide © 2004 Thomson/South-Western 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 © 2004 Thomson/South-Western Coefficient of Determination r 2 = SSR/SST = 100/114 =.8772 The regression relationship is very strong; 88% The regression relationship is very strong; 88% 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.

25 Slide © 2004 Thomson/South-Western Using Excel to Compute the Coefficient of Determination n Producing r 2 Step 3 When the Add Trendline dialog box appears: On the Options tab, select the Display On the Options tab, select the Display R-squared value on chart box R-squared value on chart box Click OK Click OK Step 2 When the Chart menu appears: Choose the Add Trendline option Choose the Add Trendline option Step 1 Position the mouse pointer over any data point in the scatter diagram and right click point in the scatter diagram and right click

29 Slide © 2004 Thomson/South-Western 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.

30 Slide © 2004 Thomson/South-Western Testing for Significance To test for a significant regression relationship, we To test for a significant regression relationship, we must conduct a hypothesis test to determine whether must conduct a hypothesis test to determine whether the value of  1 is zero. the value of  1 is zero. To test for a significant regression relationship, we To test for a significant regression relationship, we must conduct a hypothesis test to determine whether must conduct a hypothesis test to determine whether the value of  1 is zero. the value of  1 is zero. Two tests are commonly used: Two tests are commonly used: t Test and F Test Both the t test and F test require an estimate of  2, Both the t test and F test require an estimate of  2, the variance of  in the regression model. the variance of  in the regression model. Both the t test and F test require an estimate of  2, Both the t test and F test require an estimate of  2, the variance of  in the regression model. the variance of  in the regression model.

31 Slide © 2004 Thomson/South-Western An Estimate of  An Estimate of  Testing for Significance where: s 2 = MSE = SSE/( n  2) The mean square error (MSE) provides the estimate of  2, and the notation s 2 is also used.

32 Slide © 2004 Thomson/South-Western Testing for Significance An Estimate of  An Estimate of  To estimate  we take the square root of  2. To estimate  we take the square root of  2. The resulting s is called the standard error of The resulting s is called the standard error of the estimate. the estimate.

35 Slide © 2004 Thomson/South-Western 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  =.05 4. State the rejection rule. Reject H 0 if | t| > 3.182 Using the Test Statistic Using the Test Statistic Testing for Significance: t Test (3 degrees of freedom)

36 Slide © 2004 Thomson/South-Western Testing for Significance: t Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. At the.05 level of significance, the sample At the.05 level of significance, the sample evidence indicates that there is a significant relationship between the number of TV ads aired and the number of cars sold. Because t = 4.63 > 3.182, we reject H 0. Using the Test Statistic Using the Test Statistic

37 Slide © 2004 Thomson/South-Western Confidence Interval for  1 H 0 is rejected if the hypothesized value of  1 is not H 0 is rejected if the hypothesized value of  1 is not included in the confidence interval for  1. included in the confidence interval for  1. We can use a 95% confidence interval for  1 to test We can use a 95% confidence interval for  1 to test the hypotheses just used in the t test. the hypotheses just used in the t test.

38 Slide © 2004 Thomson/South-Western The form of a confidence interval for  1 is: The form of a confidence interval for  1 is: Confidence Interval for  1 where is the t value providing an area of  /2 in the upper tail of a t distribution with n - 2 degrees of freedom b 1 is the pointestimator is the margin of error

39 Slide © 2004 Thomson/South-Western n Rejection Rule 95% Confidence Interval for  1 95% Confidence Interval for  1 n Conclusion Confidence Interval for  1 Reject H 0 if 0 is not included in the confidence interval for  1. 0 is not included in the confidence interval. Reject H 0 = 5 +/- 3.182(1.08) = 5 +/- 3.44 or 1.56 to 8.44

41 Slide © 2004 Thomson/South-Western n Rejection Rule Testing for Significance: F Test where: F  is based on an F distribution with 1 degree of freedom in the numerator and n - 2 degrees of freedom in the denominator Reject H 0 if F > F 

42 Slide © 2004 Thomson/South-Western 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  =.05 4. State the rejection rule. Reject H 0 if F > 10.13 Using the Test Statistic Using the Test Statistic Testing for Significance: F Test F = MSR/MSE (1 d.f. in numerator, 3 d.f. in denominator) 3 d.f. in denominator)

43 Slide © 2004 Thomson/South-Western Testing for Significance: F Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. Because F = 21.43 > 10.13, we reject H 0. Using the Test Statistic Using the Test Statistic F = MSR/MSE = 100/4.667 = 21.43 At the.05 level of significance, the statistical At the.05 level of significance, the statistical evidence is sufficient to conclude that we have a significant relationship between the number of TV ads aired and the number of cars sold.

44 Slide © 2004 Thomson/South-Western Some Cautions about the Interpretation of Significance Tests Just because we are able to reject H 0 :  1 = 0 and Just because we are able to reject H 0 :  1 = 0 and demonstrate statistical significance does not enable demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y. Rejecting H 0 :  1 = 0 and concluding that the Rejecting H 0 :  1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y.

45 Slide © 2004 Thomson/South-Western Using Excel’s Regression Tool The Regression tool can be used to perform a The Regression tool can be used to perform a complete regression analysis. complete regression analysis. Excel also has a comprehensive tool in its Data Excel also has a comprehensive tool in its Data Analysis package called Regression. Analysis package called Regression. Up to this point, you have seen how Excel can be Up to this point, you have seen how Excel can be used for various parts of a regression analysis. used for various parts of a regression analysis.

47 Slide © 2004 Thomson/South-Western Using Excel’s Regression Tool n Performing the Regression Analysis Step 3 Choose Regression from the list of Analysis Tools Analysis Tools Step 2 Choose the Data Analysis option Step 1 Select the Tools pull-down menu

48 Slide © 2004 Thomson/South-Western Using Excel’s Regression Tool n Performing the Regression Analysis Step 4 When the Regression dialog box appears: Enter C1:C6 in the Input Y Range box Enter C1:C6 in the Input Y Range box Enter B1:B6 in the Input X Range box Enter B1:B6 in the Input X Range box Select Labels Select Labels Select Confidence Level Select Confidence Level Enter 95 in the Confidence Level box Enter 95 in the Confidence Level box Select Output Range Select Output Range Enter A9 (any cell) in the Ouput Range box Enter A9 (any cell) in the Ouput Range box Click OK to begin the regression analysis Click OK to begin the regression analysis

55 Slide © 2004 Thomson/South-Western n Confidence Interval Estimate of E ( y p ) n Prediction Interval Estimate of y p Using the Estimated Regression Equation for Estimation and Prediction where: confidence coefficient is 1 -  and t  /2 is based on a t distribution with n - 2 degrees of freedom

62 Slide © 2004 Thomson/South-Western The 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: Prediction Interval for y p 25 + 8.28 = 16.72 to 33.28 cars

63 Slide © 2004 Thomson/South-Western Residual Analysis Much of the residual analysis is based on an Much of the residual analysis is based on an examination of graphical plots. examination of graphical plots. Residual for Observation i Residual for Observation i The residuals provide the best information about . The residuals provide the best information about . If the assumptions about the error term  appear If the assumptions about the error term  appear questionable, the hypothesis tests about the questionable, the hypothesis tests about the significance of the regression relationship and the significance of the regression relationship and the interval estimation results may not be valid. interval estimation results may not be valid.

64 Slide © 2004 Thomson/South-Western Residual Plot Against x If the assumption that the variance of  is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then If the assumption that the variance of  is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then The residual plot should give an overall The residual plot should give an overall impression of a horizontal band of points impression of a horizontal band of points

69 Slide © 2004 Thomson/South-Western Using Excel’s Regression Tool to Construct a Residual Plot n Producing a Residual Plot The output will include two new items: The output will include two new items: A plot of the residuals against the A plot of the residuals against the independent variable, and independent variable, and A list of predicted values of y and the A list of predicted values of y and the corresponding residual values. corresponding residual values. When the Regression dialog box appears, we must When the Regression dialog box appears, we must also select the Residual Plot option. also select the Residual Plot option. The steps outlined earlier to obtain the regression The steps outlined earlier to obtain the regression output are performed with one change. output are performed with one change.

1 1 Slide © 2004 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University.

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1 1 Slide © 2004 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University.

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