Exam 14: Introduction to Multiple

SCENARIO 14-8 A financial analyst wanted to examine the relationship between salary (in $\$ 1,000$ ) and 2 variables: age $\left(X_{1}=\mathrm{Age}\right)$ and experience in the field $\left(X_{2}=\right.$ Exper). He took a sample of 20 employees and obtained the following Microsoft Excel output: Regression Statistics Multiple R 0.8535 R Square 0.7284 Adjusted R Square 0.6964 Standard Error 10.5630 Observations 20 $\text { ANOVA }$ Coefficients Standard Error t Stat P-value Lower 95\% O5\% Intercept 1.5740 9.2723 0.1698 0.8672 -17.9888 21.1368 Age 1.3045 0.1956 6.6678 0.0000 0.8917 1.7173 Exper -0.1478 0.1944 -0.7604 0.4574 -0.5580 0.2624 Also the sum of squares due to the regression for the model that includes only Age is 5022.0654 while the sum of squares due to the regression for the model that includes only Exper is 125.9848. -Referring to Scenario 14-8, the analyst decided to construct a 95% confidence interval for β2 . The confidence interval is from ________ to ________.

SCENARIO 14-20-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataA.XLSX. S You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-A, construct the residual plot for land size.

SCENARIO 14-20-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataA.XLSX. S You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-A, the predicted total milk production of a farm with 40 milking cows, 30 acres of land and 3 laborers is _____ liters.

SCENARIO 14-20-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataA.XLSX. S You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-A, there is sufficient evidence that at least one of the explanatory variables is related to total milk production at a 1% level of significance when testing whether there is a significant relationship between total milk production and the entire set of explanatory variables.

14-30 Introduction to Multiple Regression -Referring to Scenario 14-7, the estimate of the unit change in the mean of Y per unit change in $X _ { 1 } \text {, holding } X _ { 2 }$ constant, is ________.

When an additional explanatory variable is introduced into a multiple regression model, the coefficient of multiple determination will never decrease.

An interaction term in a multiple regression model may be used when the relationship between X1 and Y changes for differing values of $X _ { 2 }$ .

SCENARIO 14-5 A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies. She proceeds to randomly select 26 large corporations and record information in millions of dollars. The Microsoft Excel output below shows results of this multiple regression. -Referring to Scenario 14-5, what are the predicted sales (in millions of dollars) for a company spending $500 million on capital and $200 million on wages?

You have just computed a regression model in which the value of coefficient of multiple determination is 0.57. To determine if this indicates that the independent variables explain a significant portion of the variation in the dependent variable, you would perform an F- test.

SCENARIO 14-3 An economist is interested to see how consumption for an economy (in $ billions) is influenced by gross domestic product ($ billions) and aggregate price (consumer price index). The Microsoft Excel output of this regression is partially reproduced below. -Referring to Scenario 14-3, to test for the significance of the coefficient on aggregate price index, the value of the relevant t-statistic is

SCENARIO 14-11 A weight-loss clinic wants to use regression analysis to build a model for weight loss of a client (measured in pounds). Two variables thought to affect weight loss are client's length of time on the weight-loss program and time of session. These variables are described below: $Y=$ Weight loss (in pounds) $X_{1}=$ Length of time in weight-loss program (in months) $X_{2}=1$ if morning session, 0 if not Data for 25 clients on a weight-loss program at the clinic were collected and used to fit the interaction model: $Y=\beta_{0}+\beta_{1} X_{1}+\beta_{2} X_{2}+\beta_{3} X_{1} X_{2}+\varepsilon$ $\text { Output from Microsoft Excel follows: }$ Regression Statistics Multiple R 0.7308 R Square 0.5341 Adjusted R Square 0.4675 Standard Error 43.3275 Observations 25 $\text { ANOVA }$ Coefficients Standard Error t Stot Lower 99\% Upper 99\% Intercept -20.7298 22.3710 -0.9266 0.3646 -84.0702 42.6106 Length 7.2472 1.4992 4.8340 0.0001 3.0024 11.4919 Morn 90.1981 40.2336 2.2419 0.0359 -23.7176 204.1138 Length × Morn -5.1024 3.3511 -1.5226 0.1428 -14.5905 4.3857 -Referring to Scenario 14-11, which of the following statements is supported by the analysis shown? a) There is sufficient evidence (at $\alpha = 0.05$ ) of curvature in the relationship between weight loss $( Y )$ and months on program $\left( X _ { 1 } \right)$ . b) There is sufficient evidence (at $\alpha = 0.05$ ) to indicate that the relationship between weight loss $( Y )$ and months on program $\left( X _ { 1 } \right)$ varies with session time. c) There is insufficient evidence (at $\alpha = 0.05$ ) of curvature in the relationship between weight loss $( Y )$ and months on program $\left( X _ { 1 } \right)$ . d) There is insufficient evidence (at $\alpha = 0.05$ ) to indicate that the relationship between weight loss $( Y )$ and months on program $\left( X _ { 1 } \right)$ varies with session time.

SCENARIO 14-3 An economist is interested to see how consumption for an economy (in $ billions) is influenced by gross domestic product ($ billions) and aggregate price (consumer price index). The Microsoft Excel output of this regression is partially reproduced below. -Referring to Scenario 14-3, the p-value for the aggregated price index is

SCENARIO 14-8 A financial analyst wanted to examine the relationship between salary (in $\$ 1,000$ ) and 2 variables: age $\left(X_{1}=\mathrm{Age}\right)$ and experience in the field $\left(X_{2}=\right.$ Exper). He took a sample of 20 employees and obtained the following Microsoft Excel output: Regression Statistics Multiple R 0.8535 R Square 0.7284 Adjusted R Square 0.6964 Standard Error 10.5630 Observations 20 $\text { ANOVA }$ Coefficients Standard Error t Stat P-value Lower 95\% O5\% Intercept 1.5740 9.2723 0.1698 0.8672 -17.9888 21.1368 Age 1.3045 0.1956 6.6678 0.0000 0.8917 1.7173 Exper -0.1478 0.1944 -0.7604 0.4574 -0.5580 0.2624 Also the sum of squares due to the regression for the model that includes only Age is 5022.0654 while the sum of squares due to the regression for the model that includes only Exper is 125.9848. -Referring to Scenario 14-8, the p-value of the F test for the significance of the entire regression is ________.

SCENARIO 14-20-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataA.XLSX. S You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-A, construct the normal probability plot for the regression residuals.

To explain personal consumption (CONS) measured in dollars, data is collected for INC: personal income in dollars CRDTLIM: $1 plus the credit limit in dollars available to the individual APR: mean annualized percentage interest rate for borrowing for the individual ADVT: per person advertising expenditure in dollars by manufacturers in the city where The individual lives SEX: gender of the individual; 1 if female, 0 if male A regression analysis was performed with CONS as the dependent variable and CRDTLIM, APR, ADVT, and GENDER as the independent variables. The estimated model was $\hat { Y } = 2.28 - 0.29 \text { CRDTLIM + } 5.77 \text { APR + } 2.35 \text { ADVT + } 0.39 \text { SEX }$ What is the correct interpretation for the estimated coefficient for GENDER?

SCENARIO 14-20-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-B, what is the value of the coefficient of partial determination $r _ { Y 2.13 } ^ { 2 }$ ?

SCENARIO 14-20-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataA.XLSX. S You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-A, there is sufficient evidence that total milk production depends on each of the explanatory variables at a 1% level of significance when testing whether there is a significant relationship between total milk production and the entire set of explanatory variables.

SCENARIO 14-19 The marketing manager for a nationally franchised lawn service company would like to study the characteristics that differentiate home owners who do and do not have a lawn service. A random sample of 30 home owners located in a suburban area near a large city was selected; 11 did not have a lawn service (code 0) and 19 had a lawn service (code 1). Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars) and lawn size (Lawn Size, in thousands of square feet). The PHStat output is given below: Binary Logistic Regression Predictor Coefficients SE Coef Z p -Value Intercept -7.8562 3.8224 -2.0553 0.0398 Income 0.0304 0.0133 2.2897 0.0220 Lawn Size 1.2804 0.6971 1.8368 0.0662 Deviance 25.3089 -Referring to Scenario 14-19, what should be the decision ('reject' or 'do not reject') on the null hypothesis when testing whether Income makes a significant contribution to the model in the presence of LawnSize at a 0.05 level of significance?

SCENARIO 14-14 An automotive engineer would like to be able to predict automobile mileages. She believes that the two most important characteristics that affect mileage are horsepower and the number of cylinders (4 or 6) of a car. She believes that the appropriate model is Y=40-0.05+20-0.1 where = horsepower =1 if 4 cylinders, 0 if 6 cylinders Y= mileage. -Referring to Scenario 14-14, the fitted model for predicting mileages for 4-cylinder cars is ________.

SCENARIO 14-19 The marketing manager for a nationally franchised lawn service company would like to study the characteristics that differentiate home owners who do and do not have a lawn service. A random sample of 30 home owners located in a suburban area near a large city was selected; 11 did not have a lawn service (code 0) and 19 had a lawn service (code 1). Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars) and lawn size (Lawn Size, in thousands of square feet). The PHStat output is given below: Binary Logistic Regression Predictor Coefficients SE Coef Z p -Value Intercept -7.8562 3.8224 -2.0553 0.0398 Income 0.0304 0.0133 2.2897 0.0220 Lawn Size 1.2804 0.6971 1.8368 0.0662 Deviance 25.3089 -Referring to Scenario 14-19, what is the estimated odds ratio for a home owner with a family income of $100,000 and a lawn size of 2,000 square feet?