Exam 14: Introduction to Multiple

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, which of the following is a correct interpretation for the coefficient of partial determination $r _ { Y 1.23 } ^ { 2 }$ ?

SCENARIO 14-10 You worked as an intern at We Always Win Car Insurance Company last summer. You notice that individual car insurance premiums depend very much on the age of the individual and the number of traffic tickets received by the individual. You performed a regression analysis in EXCEL and obtained the following partial information: Regression Statistics Multiple R 0.8546 R Square 0.7303 Adjusted R Square 0.6853 Standard Error 226.7502 Observations 15 $\text { ANOVA }$ Coefficients Standard Error tStat P-value Lower 99\% Upper 99\% Intercept 821.2617 161.9391 5.0714 0.0003 326.6124 1315.9111 Age -1.4061 2.5988 -0.5411 0.5984 -9.3444 6.5321 Tickets 243.4401 43.2470 5.6291 0.0001 111.3406 375.5396 -Referring to Scenario 14-10, the regression sum of squares that is missing in the ANOVA table should be ______.

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 p-value of the test statistic when testing whether Income makes a significant contribution to the model in the presence of LawnSize?

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, which of the following values for α is the smallest for which the regression model as a whole is significant?

SCENARIO 14-4 A real estate builder wishes to determine how house size (House) is influenced by family income (Income) and family size (Size). House size is measured in hundreds of square feet and income is measured in thousands of dollars. The builder randomly selected 50 families and ran the multiple regression. Partial Microsoft Excel output is provided below: -Referring to Scenario 14-4, the value of the partial F test statistic is ____ for $H _ { 0 }$ : Variable $X _ { 2 }$ does not significantly improve the model after variable $X _ { l }$ has been included $H _ { 1 }$ : Variable $X _ { 2 }$ significantly improves the model after variable $X _ { 1 }$ has been included

SCENARIO 14-17 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age) and a dummy variable for management position (Manager: 1 = yes, 0 = no). The results of the regression analysis are given below: \ Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R Square 0.3765 Standard Error 18.8929 Observations 40 $\text { ANOVA }$ Coefficients Standard Error t Stat P-value Intercept -0.2143 11.5796 -0.0185 0.9853 Age 1.4448 0.3160 4.5717 0.0000 Manager -22.5761 11.3488 -1.9893 0.0541 -Referring to Scenario 14-17, the alternative hypothesis $H _ { 1 } : \text { At least one of } \beta _ { j } \neq 0 \text { for } j = 1,2$ implies that the number of weeks a worker is unemployed due to a layoff is affected by all of the 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 is the p-value of the test statistic when testing whether LawnSize makes a significant contribution to the model in the presence of Income?

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, which of the following is a correct interpretation for the slope coefficient of milking cows?

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 regression model as a whole is

SCENARIO 14-15 The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), mean teacher salary in thousands of dollars (Salaries), and instructional spending per pupil in thousands of dollars (Spending) of 47 schools in the state. Following is the multiple regression output with $Y = \%$ Passing as the dependent variable, $X _ { 1 } =$ Salaries and $X _ { 2 } =$ Spending: Regression Statistics Multiple R 0.4276 R Square 0.1828 Adjusted R Square 0.1457 Standard Error 5.7351 Observations 47 ANOVA Coefficients Standard Error t Stat \rho -value Lower 95\% Upper 95\% Intercept -72.9916 45.9106 -1.5899 0.1190 -165.5184 19.5352 Salary 2.7939 0.8974 3.1133 0.0032 0.9853 4.6025 Spending 0.3742 0.9782 0.3825 0.7039 -1.5972 2.3455 -Referring to Scenario 14-15, which of the following is the correct alternative hypothesis to test whether mean teacher salary has any effect on percentage of students passing the proficiency test, Taking into account the effect of instructional spending per pupil? a) $H _ { 1 } : \beta _ { 0 } \neq 0$ b) $H _ { 1 } : \beta _ { 1 } \neq 0$ c) $H _ { 1 } : \beta _ { 2 } \neq 0$ d) $H _ { 1 } : \beta _ { 3 } \neq 0$

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, which of the following is the correct interpretation for the Income slope coefficient?

SCENARIO 14-17 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age) and a dummy variable for management position (Manager: 1 = yes, 0 = no). The results of the regression analysis are given below: \ Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R Square 0.3765 Standard Error 18.8929 Observations 40 $\text { ANOVA }$ Coefficients Standard Error t Stat P-value Intercept -0.2143 11.5796 -0.0185 0.9853 Age 1.4448 0.3160 4.5717 0.0000 Manager -22.5761 11.3488 -1.9893 0.0541 -Referring to Scenario 14-17, the null hypothesis $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = 0$ implies that the number of weeks a worker is unemployed due to a layoff is not affected by some of the explanatory variables.

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 lower and upper limits of the 95% prediction interval for the total milk production of a farm with 40 milking cows, 30 acres of land and 3 laborers are _____ liters and _____ liters, respectively.

SCENARIO 14-17 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age) and a dummy variable for management position (Manager: 1 = yes, 0 = no). The results of the regression analysis are given below: \ Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R Square 0.3765 Standard Error 18.8929 Observations 40 $\text { ANOVA }$ Coefficients Standard Error t Stat P-value Intercept -0.2143 11.5796 -0.0185 0.9853 Age 1.4448 0.3160 4.5717 0.0000 Manager -22.5761 11.3488 -1.9893 0.0541 -Referring to Scenario 14-17, which of the following is a correct statement?

SCENARIO 14-15 The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), mean teacher salary in thousands of dollars (Salaries), and instructional spending per pupil in thousands of dollars (Spending) of 47 schools in the state. Following is the multiple regression output with $Y = \%$ Passing as the dependent variable, $X _ { 1 } =$ Salaries and $X _ { 2 } =$ Spending: Regression Statistics Multiple R 0.4276 R Square 0.1828 Adjusted R Square 0.1457 Standard Error 5.7351 Observations 47 ANOVA Coefficients Standard Error t Stat \rho -value Lower 95\% Upper 95\% Intercept -72.9916 45.9106 -1.5899 0.1190 -165.5184 19.5352 Salary 2.7939 0.8974 3.1133 0.0032 0.9853 4.6025 Spending 0.3742 0.9782 0.3825 0.7039 -1.5972 2.3455 -Referring to Scenario 14-15, the alternative hypothesis $H _ { 1 } : \text { At least one of } \beta _ { j } \neq 0 \text { for } j = 1,2$ implies that percentage of students passing the proficiency test is affected by at least one of the explanatory variables.

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 value of the F-statistic for testing the significance of the entire regression is ________.

SCENARIO 14-15 The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), mean teacher salary in thousands of dollars (Salaries), and instructional spending per pupil in thousands of dollars (Spending) of 47 schools in the state. Following is the multiple regression output with $Y = \%$ Passing as the dependent variable, $X _ { 1 } =$ Salaries and $X _ { 2 } =$ Spending: Regression Statistics Multiple R 0.4276 R Square 0.1828 Adjusted R Square 0.1457 Standard Error 5.7351 Observations 47 ANOVA Coefficients Standard Error t Stat \rho -value Lower 95\% Upper 95\% Intercept -72.9916 45.9106 -1.5899 0.1190 -165.5184 19.5352 Salary 2.7939 0.8974 3.1133 0.0032 0.9853 4.6025 Spending 0.3742 0.9782 0.3825 0.7039 -1.5972 2.3455 -Referring to Scenario 14-15, the null hypothesis $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = 0$ implies that percentage of students passing the proficiency test is not related to either of the 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 is the estimated probability that a home owner with a family income of $100,000 and a lawn size of 2,000 square feet will purchase a lawn service?

SCENARIO 14-4 A real estate builder wishes to determine how house size (House) is influenced by family income (Income) and family size (Size). House size is measured in hundreds of square feet and income is measured in thousands of dollars. The builder randomly selected 50 families and ran the multiple regression. Partial Microsoft Excel output is provided below: -Referring to Scenario 14-4, what fraction of the variability in house size is explained by income and size of family?

In a multiple regression model, the value of the coefficient of multiple determination a) has to fall between $- 1$ and $+ 1$ . b) has to fall between 0 and $+ 1$ . c) has to fall between $- 1$ and 0 . d) can fall between any pair of real numbers.