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, there is enough evidence to conclude that engine size makes a significant contribution to the regression model in the presence of the other independent variable at a 5% 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 probability that a school with a mean SAT score of 1250 and a TOEFL criterion that is at least 90?

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 partial F test for $H _ { 0 }$ : Variable $X _ { 1 }$ does not significantly improve the model after variable $X _ { 2 }$ has been included $H _ { l }$ : Variable $X _ { l }$ significantly improves the model after variable $X _ { 2 }$ has been included has $\underline{\quad\quad}$ and $\underline{\quad\quad}$ degrees of freedom.

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 -In a multiple regression model, the adjusted $r ^ { 2 }$

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 the correct null 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) $H _ { 0 } : \beta _ { 0 } = 0$ b) $H _ { 0 } : \beta _ { 1 } = 0$ c) $H _ { 0 } : \beta _ { 2 } = 0$ d) $H _ { 0 } : \beta _ { 3 } = 0$

A regression had the following results: SST = 102.55, SSE = 82.04. It can be said that 20.0% of the variation in the dependent variable is explained by the independent variables in the regression.

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 partial F test statistic is ____ for $H _ { 0 }$ : Variable $X _ { 2 }$ does not significantly improve the model after variable $X _ { I }$ has been included $H _ { 1 }$ : Variable $X _ { 2 }$ significantly improves the model after variable $X _ { 1 }$ has been included

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 wants to use a t test to test for the significance of the coefficient of $X _ { 2 }$ . At a level of significance of 0.01, the department head would decide that $\beta _ { 2 } \neq 0$ .

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 any of the explanatory variables.

In trying to construct a model to estimate grades on a statistics test, a professor wanted to include, among other factors, whether the person had taken the course previously. To do this, the professor included a dummy variable in her regression model that was equal to 1 if the person had previously taken the course, and 0 otherwise. The interpretation of the coefficient associated with this dummy variable would be the mean amount the repeat students tended to be above or below non-repeaters, with all other factors the same.

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, one individual in the sample had an annual income of $40,000 and a family size of 1. This individual owned a home with an area of 1,000 square feet (House = 10.00). What is the residual (in hundreds of square feet) for this data point?

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% confidence interval for the mean total milk production of a farm with 30 milking cows, 20 acres of land and 3 laborers are _____ liters and _____ liters, respectively.

SCENARIO 14-2 A professor of industrial relations believes that an individual's wage rate at a factory $( Y )$ depends on his performance rating $\left( X _ { 1 } \right)$ and the number of economics courses the employee successfully completed in college $\left( X _ { 2 } \right)$ . The professor randomly selects 6 workers and collects the following information: -Referring to Scenario 14-2, for these data, what is the estimated coefficient for the number of economics courses taken, $b _ { 2 }$ ?

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 estimated mean total milk production of a farm with 30 milking cows, 20 acres of land and 3 laborers is _____ liters.

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, there is not enough evidence to conclude that Toefl90 makes a significant contribution to the model in the presence of SAT at a 0.05 level of significance.

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, ________ of the variation in Accel Time can be explained by the two independent variables after taking into consideration the number of independent variables and the number of observations.

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 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 14-16 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, what is the p-value of the test statistic to determine whether engine size makes a significant contribution to the regression model in the presence of the other independent variable at a 5% level of significance?

14-30 Introduction to Multiple Regression -Referring to Scenario 14-7, the predicted GPA for a student carrying 15 course units and who has a total SAT of 1,100 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 predicted salary (in $1,000) for a 35-year-old person with 10 years of experience is ________.