1. 1 1 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2. 2 2 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 14, Part A Simple Linear Regression n Simple Linear Regression Model n Least Squares Method n Coefficient of Determination n Model Assumptions n Testing for Significance

3. 3 3 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 4 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 5 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 6 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 7 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 8 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 9 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. 10 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. 11 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

12. 12 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. 13 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. 14 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n y -Intercept for the Estimated Regression Equation Least Squares Method

15. 15 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. 16 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression n Example: Reed Auto Sales Number of TV Ads ( x ) TV Ads ( x ) Number of Cars Sold ( y ) 1 3 2 1 3 14 24 18 17 27  x = 10  y = 100

17. 17 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Estimated Regression Equation n Slope for the Estimated Regression Equation n y -Intercept for the Estimated Regression Equation n Estimated Regression Equation

18. 18 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Worksheet (showing data) Estimated Regression Equation

19. 19 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Producing a Scatter Diagram Step 1 Select cells B2:C6 Step 2 Click the Insert tab on the 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 Estimated Regression Equation Step 5 In the Chart Layouts group, click Layout 1 Step 6 Right-click on the Chart Title to display a list of options; choose Delete of options; choose Delete

20. 20 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Producing a Scatter Diagram (continued) Step 7 Select the Horizontal (Value) Axis Title and replace it with TV Ads replace it with TV Ads Step 8 Select the Vertical (Value) Axis Title and replace it with Cars Sold replace it with Cars Sold Step 9 Right-click on the Series 1 Legend Entry to display a list of options; choose Delete display a list of options; choose Delete Step 10 Position the mouse pointer over any Vertical (Value) Axis Major Gridline in the Vertical (Value) Axis Major Gridline in the scatter diagram and right-click to display a scatter diagram and right-click to display a list of options; choose Delete list of options; choose Delete Estimated Regression Equation

21. 21 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Producing a Scatter Diagram (continued) Step 11 Position the mouse pointer over any data point in the scatter diagram and right-click to point in the scatter diagram and right-click to display a list of options; choose Add Trendline display a list of options; choose Add Trendline Estimated Regression Equation Step 12 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

22. 22 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Estimated Regression Equation

23. 23 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation Reed Auto Sales Estimated Regression Line

24. 24 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

25. 25 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

26. 26 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

27. 27 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Using Excel to Produce r 2 Step 3 When the Format 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 Step 2 Choose 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 to display a list of options to display a list of options Coefficient of Determination

28. 28 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Value Worksheet (showing r 2 ) Coefficient of Determination

29. 29 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sample Correlation Coefficient where: b 1 = the slope of the estimated regression b 1 = the slope of the estimated regression equation equation

30. 30 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The sign of b 1 in the equation is “+”. Sample Correlation Coefficient r xy = +.9366

31. 31 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

32. 32 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

33. 33 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. An Estimate of  2 An Estimate of  2 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.

34. 34 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. 35 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Hypotheses n Test Statistic Testing for Significance: t Test where

36. 36 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Rejection Rule Testing for Significance: t Test where: t  is based on a t distribution with n - 2 degrees of freedom Reject H 0 if p -value <  or t t 

37. 37 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 p -value <.05 or | t| > 3.182 (with 3 degrees of freedom) Testing for Significance: t Test

38. 38 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance: t Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. t = 4.541 provides an area of.01 in the upper tail. Hence, the p -value is less than.02. (Also, t = 4.63 > 3.182.) We can reject H 0.

39. 39 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

40. 40 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

41. 41 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 n Rejection Rule 95% Confidence Interval for  1 95% Confidence Interval for  1 n Conclusion

42. 42 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Hypotheses n Test Statistic Testing for Significance: F Test F = MSR/MSE

43. 43 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 p -value <  p -value <  or F > F 

44. 44 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 p -value <.05 or F > 10.13 (with 1 d.f. in numerator and 3 d.f. in denominator) 3 d.f. in denominator) Testing for Significance: F Test F = MSR/MSE

45. 45 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance: F Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. F = 17.44 provides an area of.025 in the upper tail. Thus, the p -value corresponding to F = 21.43 is less than 2(.025) =.05. Hence, we reject H 0. F = 17.44 provides an area of.025 in the upper tail. Thus, the p -value corresponding to F = 21.43 is less than 2(.025) =.05. Hence, we reject H 0. F = MSR/MSE = 100/4.667 = 21.43 The statistical evidence is sufficient to conclude 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.

46. 46 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

47. 47 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 14, Part A