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 error appears to be left-skewed.

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 at least one of the explanatory 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, which of the following is a correct statement?

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 fraction of the variability in sales is explained by spending on capital and wages?

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 are the lower and upper limits of the 95% confidence interval estimate for the change in mean total milk production as a result of adding one more laborer after taking into consideration the effect of all the other independent 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, there is enough evidence to conclude that being a sedan or not makes a significant contribution to the regression model in the presence of the other independent variable at a 5% level of significance.

SCENARIO 14-13 An econometrician is interested in evaluating the relationship of demand for building materials to mortgage rates in Los Angeles and San Francisco. He believes that the appropriate model is $Y = 10 + 5 X _ { 1 } + 8 X _ { 2 }$ where where = mortgage rate in \% =1 if SF, 0 if LA Y= demand in \ 100 per capita -Referring to Scenario 14-13, the fitted model for predicting demand in Los Angeles is ________. a) $10 + 5 X _ { 1 }$ b) $10 + 13 X _ { 1 }$ c) $15 + 8 X _ { 2 }$ d) $18 + 5 X _ { 2 }$

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 are the degrees of freedom for the chi-square distribution when testing whether the model is a good-fitting model?

SCENARIO 14-13 An econometrician is interested in evaluating the relationship of demand for building materials to mortgage rates in Los Angeles and San Francisco. He believes that the appropriate model is $Y = 10 + 5 X _ { 1 } + 8 X _ { 2 }$ where where = mortgage rate in \% =1 if SF, 0 if LA Y= demand in \ 100 per capita -Referring to Scenario 14-13, holding constant the effect of city, each additional increase of 1% in the mortgage rate would lead to an estimated increase of ________ in the mean demand.

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 Toefl90 slope coefficient?

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 coefficient of partial determination $r _ { Y 1 \cdot 2 } ^ { 2 }$ ⋅ is ____.

14-30 Introduction to Multiple Regression -Referring to Scenario 14-7, the net regression coefficient of $X _ { 2 }$ is ________.

SCENARIO 14-12 As a project for his business statistics class, a student examined the factors that determined parking meter rates throughout the campus area. Data were collected for the price (\$) per hour of parking, blocks to the quadrangle, and whether the parking is on or off campus. The population regression model hypothesized is $Y _ { i } = \alpha + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + \varepsilon$ where $Y$ is the meter price per hour $X _ { l }$ is the number of blocks to the quad $X _ { 2 }$ is a dummy variable that takes the value 1 if the meter is located on campus and 0 otherwise The following Excel results are obtained. Regression Statistics Multiple R 0.5536 R Square 0.3064 Adjusted R Square 0.2812 Standard Error 0.4492 Observations 58 ANOVA df SS MS F Significance F Regression 2 4.9035 2.4518 12.1501 0.0000 Residual 55 11.0984 0.2018 Total 57 16.0019 Coefficients Standard Error Stat P-value Lower 99\% Upper 99\% Intercept 1.6500 0.2028 8.1359 0.0000 1.1089 2.1912 Block -0.2504 0.0529 -4.7355 0.0000 -0.3915 -0.1093 Campus 0.1552 0.1297 1.1966 0.2366 -0.1908 0.5011 -Referring to Scenario 14-12, if one is already off campus but decides to park 3 more blocks from the quad, the estimated mean parking meter rate will decrease by ____.

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 coefficient of multiple determination is ________.

14-30 Introduction to Multiple Regression -Referring to Scenario 14-7, the department head wants to test $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = 0$ . The value of the F-test statistic 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, there is not enough evidence to conclude that Income makes a significant contribution to the model in the presence of LawnSize at a 0.05 level of significance.

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 partial F test 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 $X _ { I }$ has been included has $\underline{\quad\quad}$ and $\underline{\quad\quad}$ degrees of freedom.

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, we can conclude definitively that, holding constant the effect of the other independent variable, age has an impact on the mean number of weeks a worker is unemployed due to a layoff at a 10% level of significance if all we have is the information of the 95% confidence interval estimate for the effect of a one year increase in age on the mean number of weeks a worker is unemployed due to a layoff.

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, one economy in the sample had an aggregate consumption level of $4 billion, a GDP of $6 billion, and an aggregate price level of 200. What is the residual for this data Point?

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, to test the significance of the multiple regression model, the value of the test statistic is ______.