Exam 14: Introduction to Multiple

SCENARIO 14-16 What are the factors that determine the acceleration time (in sec.) from 0 to 60 miles per hour of a car? Data on the following variables for 30 different vehicle models were collected: $Y$ (Accel Time): Acceleration time in sec. $X _ { I }$ (Engine Size): c.c. $X _ { 2 }$ (Sedan): 1 if the vehicle model is a sedan and 0 otherwise The regression results using acceleration time as the dependent variable and the remaining variables as the independent variables are presented below. Regression Statistics Multiple R 0.6096 R Square 0.3716 Adjusted R Square 0.3251 Standard Error 1.4629 Observations 30 ANOVA Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 7.1052 0.6574 10.8086 0.0000 5.7564 8.4540 Engine Size -0.0005 0.0001 -3.6477 0.0011 -0.0008 -0.0002 Sedan 0.7264 0.5564 1.3056 0.2027 -0.4152 1.8681 -Referring to Scenario 14-16, the 0 to 60 miles per hour acceleration time of a sedan is predicted to be 0.0005 seconds lower than that of a non-sedan with the same engine size.

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, the null hypothesis should be rejected at a 10% level of significance when testing whether the number of laborers has any effect on the total milk production while holding constant the effect of the other independent variables.

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, what is the p-value of the test statistic to determine whether there is a significant relationship between percentage of students passing the proficiency test and the entire set of explanatory variables?

SCENARIO 14-1 A manager of a product sales group believes the number of sales made by an employee $( Y )$ depends on how many years that employee has been with the company $\left( X _ { 1 } \right)$ and how he/she scored on a business aptitude test $\left( X _ { 2 } \right)$ . A random sample of 8 employees provides the following: -Referring to Scenario 14-1, for these data, what is the estimated coefficient for the variable representing years an employee has been with the company, b1?

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 determine whether there is a significant relationship between percentage of students passing the Proficiency test and the entire set of explanatory variables? a) $H _ { 1 } : \beta _ { 0 } = \beta _ { 1 } = \beta _ { 2 } \neq 0$ b) $H _ { 1 } : \beta _ { 1 } = \beta _ { 2 } \neq 0$ c) $H _ { 1 } :$ At least one of $\beta _ { j } \neq 0$ for $j = 0,1,2$ d) $H _ { 1 } :$ At least one of $\beta _ { j } \neq 0$ for $j = 1,2$

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 a correct statement?

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, what is the p-value of the test statistic to determine whether there is a significant relationship between total milk production and the entire set of explanatory variables?

SCENARIO 14-18 A logistic regression model was estimated in order to predict the probability that a randomly chosen university or college would be a private university using information on mean total Scholastic Aptitude Test score (SAT) at the university or college and whether the TOEFL criterion is at least 90 (Toefl90 = 1 if yes, 0 otherwise.) The dependent variable, Y, is school type (Type = 1 if private and 0 otherwise). The PHStat output is given below: Binary Logistic Regression Predictor Coefficients SE Coef Z p -Value Intercept -3.9594 1.6741 -2.3650 0.0180 SAT 0.0028 0.0011 2.5459 0.0109 Toefl90:1 0.1928 0.5827 0.3309 0.7407 Deviance 101.9826 -Referring to Scenario 14-18, which of the following is the correct interpretation for the SAT slope coefficient?

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 are the regression degrees of freedom that are missing from the output?

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, there is sufficient evidence that the number of weeks a worker is unemployed due to a layoff depends on all of the explanatory variables at a 10% level of significance.

SCENARIO 14-18 A logistic regression model was estimated in order to predict the probability that a randomly chosen university or college would be a private university using information on mean total Scholastic Aptitude Test score (SAT) at the university or college and whether the TOEFL criterion is at least 90 (Toefl90 = 1 if yes, 0 otherwise.) The dependent variable, Y, is school type (Type = 1 if private and 0 otherwise). The PHStat output is given below: Binary Logistic Regression Predictor Coefficients SE Coef Z p -Value Intercept -3.9594 1.6741 -2.3650 0.0180 SAT 0.0028 0.0011 2.5459 0.0109 Toefl90:1 0.1928 0.5827 0.3309 0.7407 Deviance 101.9826 -Referring to Scenario 14-18, what is the estimated odds ratio for a school with a mean SAT score of 1250 and a TOEFL criterion that is at least 90?

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, there is sufficient evidence that at least one of the explanatory variables is related to the percentage of students passing the proficiency test at a 5% level of significance.

14-22 Introduction to Multiple Regression One of the most common questions of prospective house buyers pertains to the cost of heating in dollars $( Y )$ . To provide its customers with information on that matter, a large real estate firm used the following 2 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit $\left( X _ { 1 } \right)$ and the amount of insulation in inches $\left( X _ { 2 } \right)$ . Given below is EXCEL output of the regression model. Regression Statistics Multiple R 0.5270 R Square 0.2778 Adjusted R Square 0.1928 Standard Error 40.9107 Observations 20 ANOVA Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 448.2925 90.7853 4.9379 0.0001 256.7522 639.8328 Temperature -2.7621 1.2371 -2.2327 0.0393 -5.3721 -0.1520 Insulation -15.9408 10.0638 -1.5840 0.1316 -37.1736 Also $\operatorname { SSR } \left( X _ { 1 } \mid X _ { 2 } \right) = 8343.3572$ and $\operatorname { SSR } \left( X _ { 2 } \mid X _ { 1 } \right) = 4199.2672$ -Referring to Scenario 14-6, what can we say about the regression model?

14-22 Introduction to Multiple Regression One of the most common questions of prospective house buyers pertains to the cost of heating in dollars $( Y )$ . To provide its customers with information on that matter, a large real estate firm used the following 2 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit $\left( X _ { 1 } \right)$ and the amount of insulation in inches $\left( X _ { 2 } \right)$ . Given below is EXCEL output of the regression model. Regression Statistics Multiple R 0.5270 R Square 0.2778 Adjusted R Square 0.1928 Standard Error 40.9107 Observations 20 ANOVA Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 448.2925 90.7853 4.9379 0.0001 256.7522 639.8328 Temperature -2.7621 1.2371 -2.2327 0.0393 -5.3721 -0.1520 Insulation -15.9408 10.0638 -1.5840 0.1316 -37.1736 Also $\operatorname { SSR } \left( X _ { 1 } \mid X _ { 2 } \right) = 8343.3572$ and $\operatorname { SSR } \left( X _ { 2 } \mid X _ { 1 } \right) = 4199.2672$ -Referring to Scenario 14-6, the estimated value of the regression parameter $\beta _ { 1 }$ in means that

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 99% confidence interval for the change in mean insurance premiums of a person who has become 1 year older (i.e., the slope coefficient for AGE) is -1.4061 ± _______.

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, 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, construct the residual plot for the number of milking cows.

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, one company in the sample had sales of $21.439 billion (Sales = 21,439). This company spent $300 million on capital and $700 million on wages. What is the Residual (in millions of dollars) for this data point?

SCENARIO 14-1 A manager of a product sales group believes the number of sales made by an employee $( Y )$ depends on how many years that employee has been with the company $\left( X _ { 1 } \right)$ and how he/she scored on a business aptitude test $\left( X _ { 2 } \right)$ . A random sample of 8 employees provides the following: -Referring to Scenario 14-1, if an employee who had been with the company 5 years scored a 9 on the aptitude test, what would his estimated expected sales be?

14-22 Introduction to Multiple Regression One of the most common questions of prospective house buyers pertains to the cost of heating in dollars $( Y )$ . To provide its customers with information on that matter, a large real estate firm used the following 2 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit $\left( X _ { 1 } \right)$ and the amount of insulation in inches $\left( X _ { 2 } \right)$ . Given below is EXCEL output of the regression model. Regression Statistics Multiple R 0.5270 R Square 0.2778 Adjusted R Square 0.1928 Standard Error 40.9107 Observations 20 ANOVA Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 448.2925 90.7853 4.9379 0.0001 256.7522 639.8328 Temperature -2.7621 1.2371 -2.2327 0.0393 -5.3721 -0.1520 Insulation -15.9408 10.0638 -1.5840 0.1316 -37.1736 Also $\operatorname { SSR } \left( X _ { 1 } \mid X _ { 2 } \right) = 8343.3572$ and $\operatorname { SSR } \left( X _ { 2 } \mid X _ { 1 } \right) = 4199.2672$ -Referring to Scenario 14-6, the coefficient of partial determination $r _ { Y 1 \cdot 2 } ^ { 2 }$ ⋅ is ____.