© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 1 Slides by John Loucks St. Edward’s University

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 2 2 Chapter 12 Simple Linear Regression n n Simple Linear Regression Model n n Least Squares Method n n Coefficient of Determination n n Model Assumptions n n Testing for Significance n n Excel’s Regression Tool

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3 3 Chapter 12 Simple Linear Regression We will also use Excel extensively in this chapter. This chapter summary and “Show Me Solution” videos in MindTap show manual procedures which you need to understand. But you can answer all the questions in practice, quiz and assignment with Excel. “Excel’s Regression Tool” section in MindTap explains how to use Excel to do those procedures. I made an Excel video that shows how to use Excel output to do those procedures more easily, quickly and accurately. Please watch the Excel video until you are familiar with it. And use Excel whenever possible.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 4 Simple Linear Regression Regression analysis can be used to develop an equation showing how the variables are related. Managerial decisions often are based on the relationship between two or more variables. The variables being used to predict the value of the dependent variable are called the independent variables and are denoted by x. The variable being predicted is called the dependent variable and is denoted by y.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 5 The relationship between the two variables is approximated by a straight line. Simple linear regression involves one independent variable and one dependent variable. Regression analysis involving two or more independent variables is called multiple regression. Simple Linear Regression

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 6 6 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. The simple linear regression model is: The equation that describes how y is related to x and an error term is called the regression model.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 7 Simple Linear Regression Equation n n The simple linear regression equation is: E(y) is the expected value of y for a given x value.  1 is the slope of the regression line.  0 is the y intercept of the regression line. Graph of the regression equation is a straight line. E(y) =  0 +  1 x

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 8 Simple Linear Regression Equation n n Positive Linear Relationship E(y)E(y) x Slope  1 is positive Regression line Intercept  0

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 9 Simple Linear Regression Equation n n Negative Linear Relationship E(y)E(y) x Slope  1 is negative Regression line Intercept  0

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 10 Simple Linear Regression Equation n n No Relationship E(y)E(y) x Slope  1 is 0 Regression line Intercept  0

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 11 Estimated Simple Linear Regression Equation n n The estimated simple linear regression equation b 1 is the slope of the line. b 0 is the y intercept of the line. The graph is called the estimated regression line.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 12 Estimation Process Regression Model y =  0 +  1 x +  Regression Equation E(y) =  0 +  1 x Unknown Parameters  0,  1 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 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 Estimated Regression Equation Sample Statistics b 0, b 1

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 13 Least Squares Method n Least Squares Criterion where: y i = observed value of the dependent variable for the i th observation

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 14 Least Squares Method n Slope for the Estimated Regression Equation where: x i = value of independent variable for i th observation y i = value of dependent variable for i th observation

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 15 n n y-Intercept for the Estimated Regression Equation Least Squares Method

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 16 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 n Example: Reed Auto Sales

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 17 Simple Linear Regression n n Example: Reed Auto Sales Number of TV Ads (x) Number of Cars Sold (y)  x = 10  y = 100

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 18 Estimated Regression Equation n n Slope for the Estimated Regression Equation n n y-Intercept for the Estimated Regression Equation n n Estimated Regression Equation

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 19 n n Excel Worksheet (showing data) Estimated Regression Equation

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 20 n 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 Insert Scatter (X,Y) Step 4 When the list of scatter diagram subtypes appears, Click Scatter (chart in upper left corner) Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 21 n n Editing a Scatter Diagram Step 1 Click the Chart Title and replace it with Reed Auto Sales Estimated Regression Line Step 3 When the list of chart elements appears: Click Axis Titles (creates placeholders for titles) Click Gridlines (to deselect gridlines option) Click Trendline Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation Step 2 Click the Chart Elements button

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 22 n n Editing a Scatter Diagram (continued) Step 4 Click the Horizontal (Category) Axis Title and replace it with TV Ads Step 5 Click the Vertical (Value) Axis Title and replace it with Cars Sold Step 6 Select the Format Trendline option Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation Step 7 When the Format Trendline dialog box appears: Select Display equation on chart Click the Fill & Line button In the Dash type box, select Solid Close the Format Trendline dialog box

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 23 Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation Reed Auto Sales Estimated Regression Line

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 24 Coefficient of Determination n Relationship Among SST, SSR, SSE where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error SST = SSR + SSE

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 25 n 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

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 26 Coefficient of Determination r 2 = SSR/SST = 100/114 =.8772 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.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 27 Using Excel to Compute the Coefficient of Determination n n Adding r 2 Value to Scatter Diagram Step 2 When the Format Trendline dialog box appears: Select Display R-squared on chart Close the Format Trendline dialog box Step 1 Right-click on the trendline and select the Format Trendline option

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 28 n n Adding r 2 Value to Scatter Diagram Using Excel to Compute the Coefficient of Determination Reed Auto Sales Estimated Regression Line

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 29 Sample Correlation Coefficient where: b 1 = the slope of the estimated regression equation

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 30 Sample Correlation Coefficient r xy =

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 31 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. 3. The values of  are independent. 4. The error  is a normally distributed random variable.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 32 Testing for Significance To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of  1 is zero. Two tests are commonly used: t Test and F Test Both the t test and F test require an estimate of  2, the variance of e in the regression model.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 33 Testing for Significance n An Estimate of  2 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.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 34 Testing for Significance n An Estimate of s To estimate s, we take the square root of s 2. The resulting s is called the standard error of the estimate.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 35 Testing for Significance: t Test n Hypotheses n Test Statistic where H 0 :  1 = 0 H a :  1 ≠ 0

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 36 n 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 

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if p-value <.05 or |t| > (with 3 degrees of freedom) Testing for Significance: t Test H 0 :  1 = 0 H a :  1 ≠ 0

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 38 Testing for Significance: t Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. t = provides an area of.01 in the upper tail. Hence, the p-value is less than.02. (Also, t = 4.63 > ) We can reject H 0.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 39 Confidence Interval for  1 H 0 is rejected if the hypothesized value of  1 is not included in the confidence interval for  1. We can use a 95% confidence interval for  1 to test the hypotheses just used in the t test.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 40 Confidence Interval for  1 n The form of a confidence interval for  1 is: where t  /2 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 point estimator b 1 is the point estimator is the margin of error is the margin of error

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 41 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 or 1.56 to 8.44 n n Rejection Rule n n 95% Confidence Interval for  1 n n Conclusion = 5 +/ (1.08) = 5 +/- 3.44

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 42 n n Hypotheses n n Test Statistic Testing for Significance: F Test F = MSR/MSE H 0 : b 1 = 0 H a : b 1 ≠ 0

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 43 n 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 <  or F > F 

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if p-value <.05 or F > (with 1 d.f. in numerator and 3 d.f. in denominator) Testing for Significance: F Test F = MSR/MSE H 0 : b 1 = 0 H a : b 1 ≠ 0

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 45 Testing for Significance: F Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. F = provides an area of.025 in the upper tail. Thus, the p-value corresponding to F = is less than.025. Hence, we reject H 0. F = MSR/MSE = 100/4.667 = 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.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 46 Some Cautions about the Interpretation of Significance Tests Just because we are able to reject H 0 :  1 = 0 and 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 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.

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 47 Using Excel’s Regression Tool n n Performing the Regression Analysis Step 3 Choose Regression from the list of Analysis Tools Step 2 In the Analysis group, click Data Analysis Step 1 Click the DATA tab on the ribbon continued

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 48 Using Excel’s Regression Tool n n Performing the Regression Analysis Enter C1:C6 in the Input Y Range box Step 4 When the Regression dialog box appears: Enter B1:B6 in the Input X Range box Select Labels Select Confidence Level Enter 95 in the Confidence Level box Select Output Range Enter A8 in the Output Range box Click OK

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 49 Using Excel’s Regression Tool n n Excel Value Worksheet ANOVA Output Regression Statistics Output Data Estimated Regression Equation Output ABCDEFGHI 1WeekTV AdsCars Sold SUMMARY OUTPUT 9 10Regression Statistics 11Multiple R R Square Adjusted R Square Standard Error Observations ANOVA 18dfSSMSFSignificance F 19Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0% 24Intercept TV Ads

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 50 Using Excel’s Regression Tool Note: Columns F-I are not shown. n n Excel Value Worksheet (bottom-left portion)

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 51 Using Excel’s Regression Tool Note: Columns C-E are hidden. n n Excel Value Worksheet (bottom-right portion)

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 52 Using Excel’s Regression Tool n n Excel Value Worksheet (middle portion)

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 53 Using Excel’s Regression Tool n n Excel Value Worksheet (top portion)

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 54 End of Chapter 12