Exam 18: Multiple Regression

Perform statistical inference for multiple regression. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. Output is shown Below. The calculated t-statistic to determine if amount spent on advertising is a Significant independent variable in explaining Sony Bravia sales is $\text {Dependent Variable is Sales}$ Predictor Coef SE Coef T P Constant 90.19 25.08 3.60 0.001 Price -0.03055 0.01005 -3.04 0.005 Advertising 3.0926 0.3680 8.40 0.000 $S = 10.6075 \quad \mathrm { R } - \mathrm { Sq } = 84.48 \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { adj } \mathrm { j } ) = 83.3 \%$

Interpret multiple regression output. -In determining the best companies to work for, a number of variables are considered, Including size, average annual pay, and turnover rate, etc. Employee surveys are often Conducted in order to assess aspects of the organization's culture, such as trust and Openness to change. A sample of 33 companies was randomly selected and data Collected on the average annual bonus and turnover rate (%). A questionnaire was also Administered to the employees of each company to arrive at a trust index (measured on a Scale of 0 - 100). Based on the output shown below, how much of the variability in Turnover Rate is explained by the estimated multiple regression model? $\text {Dependent Variable is Turnover Rate}$ Predictor Coef SE Coef T P Constant 12.1005 0.7826 15.46 0.000 Trust Index -0.07149 0.01966 -3.64 0.001 Average Bonus -0.0007216 0.0001481 -4.87 0.000 $S = 1.49746 \quad \mathrm { R } - \mathrm { Sq } = 79.68 \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { adj } ) = 78.3\%$

Interpret multiple regression output. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. Output is shown Below. Using the estimated multiple regression model, the number of units sold on Average at a store that sells the Sony Bravia for $2199 and spends 10% of its advertising Budget on the product is $\text {Dependent Variable is Sales}$ Predictor Coef SE Coef T P Constant 90.19 25.08 3.60 0.001 Price -0.03055 0.01005 -3.04 0.005 Advertising 3.0926 0.3680 8.40 0.000 $S = 10.6075 \quad R - S q = 84.4 \% \quad R - S q ( a d j ) = 83.3 \%$

Write out the multiple regression model. -In determining the best companies to work for, a number of variables are considered, Including size, average annual pay, and turnover rate, etc. Employee surveys are often Conducted in order to assess aspects of the organization's culture, such as trust and Openness to change. A sample of 33 companies was randomly selected and data Collected on the average annual bonus and turnover rate (%). A questionnaire was also Administered to the employees of each company to arrive at a trust index (measured on a Scale of 0 - 100). Based on the output shown below, the estimated multiple regression Model is $\text {Dependent Variable is Turnover Rate}$ Predictor Coef SE Coef T P Constant 12.1005 0.7826 15.46 0.000 Trust Index -0.07149 0.01966 -3.64 0.001 Average Bonus -0.0007216 0.0001481 -4.87 0.000 $S = 1.49746 \quad R - S q = 79.6 \% \quad R - S q ( a d j ) = 78.3 \%$

Interpret multiple regression coefficients. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. Based on the results Shown below, the correct interpretation of the regression coefficient for Advertising is Dependent Variable is Sales Predictor Coef SE Coef T P Constant 90.19 25.08 3.60 0.001 Price -0.03055 0.01005 -3.04 0.005 Advertising 3.0926 0.3680 8.40 0.000

Perform statistical inference for multiple regression. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. Output is shown Below. The correct null and alternative hypotheses for testing the regression coefficient Of Price is Dependent variable is Sales Predictor Coef SE Coef T P Constant 90.19 25.08 3.60 0.001 Price -0.03055 0.01005 -3.04 0.005 Advertising 3.0926 0.3680 8.40 0.000 $S = 10.6075 \quad \mathrm { R } - \mathrm { Sq } = 84.4 \% \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { adj } \mathrm { j } ) = 83.3 \%$ A. $H _ { 0 : } : \beta _ { \mathrm { P } } \neq 0$ vs. $\mathrm { H } _ { \mathrm { A } : } \beta _ { \mathrm { P } } = 0$ B. $\mathrm { H } _ { 0 : } \beta _ { \mathrm { P } } \geq 0$ vs. $\mathrm { H } _ { \mathrm { A } } : \beta _ { \mathrm { P } } < 0$ C. $\mathrm { H } _ { 0 } : \beta _ { \mathrm { P } } \leq 0$ vs. $\mathrm { H } _ { \mathrm { A } } : \beta _ { \mathrm { P } } > 0$ D. $\mathrm { H } _ { 0 : } \beta _ { \mathrm { P } } = 0$ vs. $\mathrm { H } _ { \mathrm { A } : } \beta _ { \mathrm { P } } \neq 0$ E. $\mathrm { H } _ { 0 }$ : The regression is not significant vs. $\mathrm { H } _ { \mathrm { A } }$ : The regression is significant.

Perform statistical inference for multiple regression. -In determining the best companies to work for, a number of variables are considered, Including size, average annual pay, and turnover rate, etc. Employee surveys are often Conducted in order to assess aspects of the organization's culture, such as trust and Openness to change. A sample of 33 companies was randomly selected and data Collected on the average annual bonus and turnover rate (%). A questionnaire was also Administered to the employees of each company to arrive at a trust index (measured on a Scale of 0 - 100). Based on the output shown below, the correct interpretation of the Regression coefficient associated with Average Bonus is Dependent Variable is Turnover Rate Predictor Coef SE Coef T P Constant 12.1005 0.7826 15.46 0.000 Trust Index -0.07149 0.01966 -3.64 0.001 Average Bonus -0.0007216 0.0001481 -4.87 0.000 $S = 1.49746 \quad \mathrm { R } - \mathrm { Sq } = 79.6 \% \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { adj } \mathrm { j } ) = 78.3 \%$

Perform statistical inference for multiple regression. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. Output is shown Below. Which of the following statements is / are true? Dependent Variable is Sales Predictor Coef SE Coef T P Constant 90.19 25.08 3.60 0.001 Price -0.03055 0.01005 -3.04 0.005 Advertising 3.0926 0.3680 8.40 0.000 $S = 10.6075 \quad \mathrm { R } - \mathrm { Sq } = 84.48 \quad \mathrm { R } - \mathrm { Sq } ( \operatorname { adj } \mathrm { j } ) = 83.3\%$ Analysis of Variance Source DF SS MS Regression 2 16477.3 8238.7 Residual Error 27 3038.0 112.5 Total 29 19515.4

Perform statistical inference for multiple regression. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. Output is shown Below. The calculated F statistic to determine the overall significance of the estimated Multiple regression model is Analysis of Variance Source DF SS MS Regression 2 16477.3 8238.7 Residual Error 27 3038.0 112.5 Total 29 19515.4

Interpret multiple regression output. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. Output is shown Below. Based on the output shown below, how much of the variability in Sales is Explained by the estimated multiple regression model? Analysis of Variance Source DF SS MS Regression 2 16477.3 8238.7 Residual Error 27 3038.0 112.5 Total 29 19515.4

Interpret multiple regression coefficients. -In determining the best companies to work for, a number of variables are considered, Including size, average annual pay, and turnover rate, etc. Employee surveys are often Conducted in order to assess aspects of the organization's culture, such as trust and Openness to change. A sample of 33 companies was randomly selected and data Collected on the average annual bonus and turnover rate (%). A questionnaire was also Administered to the employees of each company to arrive at a trust index (measured on a Scale of 0 - 100). Based on the results shown below, the correct interpretation of the Regression coefficient for the variable Trust Index is Dependent Variable is Turnover Rate Predictor Coef SE Coef Constant 12.1005 0.7826 15.46 0.000 Trust Index -0.07149 0.01966 -3.64 0.001 Average Bonus -0.0007216 0.0001481 -4.87 0.000

8.3. Check assumptions and conditions for the multiple regression model. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. A multiple regression Model was fit to the data and the plot of residuals versus predicted values is shown Below. What does the residual plot suggest?

Interpret multiple regression output. -In determining the best companies to work for, a number of variables are considered, Including size, average annual pay, and turnover rate, etc. Employee surveys are often Conducted in order to assess aspects of the organization's culture, such as trust and Openness to change. A sample of 33 companies was randomly selected and data Collected on the average annual bonus and turnover rate (%). A questionnaire was also Administered to the employees of each company to arrive at a trust index (measured on a Scale of 0 - 100). Based on the output shown below, a company having a trust index Score of 70 and an average annual bonus of $6500 has a predicted turnover rate of $\text {Dependent Variable is Turnover Rate}$ Predictor Coef SE Coef T P Constant 12.1005 0.7826 15.46 0.000 Trust Index -0.07149 0.01966 -3.64 0.001 Average Bonus -0.0007216 0.0001481 -4.87 0.000 $S = 1.49746 \quad R - S q = 79.6 \% \quad R - S q ( a d j ) = 78.3 \%$

Interpret multiple regression coefficients. -What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were Obtained for last quarter from a sample of 30 different stores. Also collected were data On the selling price and amount spent on advertising the Sony Bravia (as a percentage of Total advertising expenditure in the previous quarter) at each store. Based on the results Shown below, the correct interpretation of the regression coefficient for Price is Dependent Variable is Sales Predictor Coef SE Coef T P Constant 90.19 25.08 3.60 0.001 Price -0.03055 0.01005 -3.04 0.005 Advertising 3.0926 0.3680 8.40 0.000

What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a variety of outlets such as large and small electronics stores, department stores, large discount chains and online. Sales figures (number of units) for the popular Sony Bravia were obtained for last quarter from a sample of 30 different stores. Also collected were data on the selling price and amount spent on advertising the Sony Bravia (as a percentage of total advertising expenditure in the previous quarter) at each store. Below are the multiple regression results. $\text {Dependent Variable is Sales}$ Predictor Coef SE Coef T P Constant 90.19 25.08 3.60 0.001 Price -0.03055 0.01005 -3.04 0.005 Advertising 3.0926 0.3680 8.40 0.000 $S = 10.6075 \quad \mathrm { R } - S q = 84.48 \quad \mathrm { R } - S q ( a d j ) = 83.3\%$ $\text {Analysis of Variance}$ Source DF SS MS Regression 2 16477.3 8238.7 Residual Error 27 3038.0 112.5 Total 29 19515.4 a. Write out the estimated regression equation. b. Is the regression equation significant overall? Explain. c. How much of the variability in Sales is explained by the regression equation? d. State the hypotheses for testing the regression coefficient of Price. Based on the results, what do you conclude? e. State the hypotheses for testing the regression coefficient of Advertising Expenditure. Based on the results, what do you conclude? f. Predict the sales for a store that sells the Sony Bravia for $2199 and spends 10% of its advertising budget on the product. g. Comment on whether the conditions for multiple regression are satisfied based on the plots shown below.