Exam 14: Introduction to Multiple Regression

TABLE 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 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 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 0.8013 R Square 0.6421 Adjusted R Square 0.6313 Standard Error 1.0507 Observations 171 $\text { ANOVA }$ df SS MS F Significance F Regression 5 326.8700 65.3740 59.2168 0.0000 Residual 165 182.1564 1.1040 Total 170 509.0263 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 12.8627 1.0927 11.7713 0.0000 10.7052 15.0202 Cargo Vol 0.0259 0.0102 2.5518 0.0116 0.0059 0.0460 HP -0.0200 0.0018 -11.3307 0.0000 -0.0235 -0.0165 MPG -0.0620 0.0303 -2.0464 0.0423 -0.1218 -0.0022 SUV 0.7679 0.4314 1.7802 0.0769 -0.0838 1.6196 Sedan 0.6427 0.2790 2.3034 0.0225 0.0918 1.1935 The various residual plots are as shown below. The coefficients of partial determination $\left( R ^ { 2 }_{Y j} \right.$ . (All variables except $\left. j \right)$ of each of the 5 predictors are, respectively, $0.0380,0.4376,0.0248,0.0188$ , and $0.0312$ . The coefficient of multiple determination for the regression model using each of the 5 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables $\left( R _ { j } ^ { 2 } \right)$ are, respectively, $0.7461,0.5676,0.6764,0.8582,0.6632$ . -Referring to 14-16, what is the value of the test statistic to determine whether SUV makes a significant contribution to the regression model in the presence of the other independent variables at a 5% level of significance?

TABLE 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), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. The coefficients of partial determination ( 2 Yj. (Allvariables except $j$ ) ) of each of the 6 predictors are, respectively, 0.2807, 0.0386, 0.0317, 0.0141, 0.0958, and 0.1201. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error 40 Observations $\text { ANOVA }$ df SS MS F significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 $\text { ANOVA }$ df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 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 Table 14-17 Model 1, what is the p-value of the test statistic when testing whether age has any effect on the number of weeks a worker is unemployed due to a layoff while holding constant the effect of all the other independent variables?

TABLE 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; 15 did not have a lawn service (code 0) and 15 had a lawn service (code 1). Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars), lawn size (Lawn Size, in thousands of square feet), attitude toward outdoor recreational activities (Atitude 0 = unfavorable, 1 = favorable), number of teenagers in the household (Teenager), and age of the head of the household (Age). The Minitab output is given below: Logistic Regression Table Odds 95\% CI Predictor Coef SE Coef Z P Ratio Lower Upper Constant -70.49 47.22 -1.49 0.135 Income 0.2868 0.1523 1.88 0.060 1.33 0.99 1.80 LawnSiz 1.0647 0.7472 1.42 0.154 2.90 0.67 12.54 Attitude -12.744 9.455 -1.35 0.178 0.00 0.00 326.06 Teenager -0.200 1.061 -0.19 0.850 0.82 0.10 6.56 Age 1.0792 0.8783 1.23 0.219 2.94 0.53 16.45 Log-Likelihood $= - 4.890$ Test that all slopes are zero: $\mathrm { G } = 31.808 , \mathrm { DF } = 5 , p$ -value $= 0.000$ Goodness-of-Fit Tests Method Chi-Square DF Pearson 9.313 24 0.997 Deviance 9.780 24 0.995 Hosmer-Lemeshow 0.571 8 1.000 -Referring to Table 14-19, there is not enough evidence to conclude that LawnSize makes a significant contribution to the model in the presence of the other independent variables at a 0.05 level of significance.

TABLE 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 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 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 0.8013 R Square 0.6421 Adjusted R Square 0.6313 Standard Error 1.0507 Observations 171 $\text { ANOVA }$ df SS MS F Significance F Regression 5 326.8700 65.3740 59.2168 0.0000 Residual 165 182.1564 1.1040 Total 170 509.0263 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 12.8627 1.0927 11.7713 0.0000 10.7052 15.0202 Cargo Vol 0.0259 0.0102 2.5518 0.0116 0.0059 0.0460 HP -0.0200 0.0018 -11.3307 0.0000 -0.0235 -0.0165 MPG -0.0620 0.0303 -2.0464 0.0423 -0.1218 -0.0022 SUV 0.7679 0.4314 1.7802 0.0769 -0.0838 1.6196 Sedan 0.6427 0.2790 2.3034 0.0225 0.0918 1.1935 The various residual plots are as shown below. The coefficients of partial determination $\left( R ^ { 2 }_{Y j} \right.$ . (All variables except $\left. j \right)$ of each of the 5 predictors are, respectively, $0.0380,0.4376,0.0248,0.0188$ , and $0.0312$ . The coefficient of multiple determination for the regression model using each of the 5 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables $\left( R _ { j } ^ { 2 } \right)$ are, respectively, $0.7461,0.5676,0.6764,0.8582,0.6632$ . -Referring to 14-16, there is enough evidence to conclude that Cargo Vol makes a significant contribution to the regression model in the presence of the other independent variables at a 5% level of significance.

TABLE 14-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and 4 variables: age (X₁ = Age), experience in the field (X₂ = Exper), number of degrees (X₃ = Degrees), and number of previous jobs in the field (X₄ = Prevjobs). He took a sample of 20 employees and obtained the following Microsoft Excel output: SUMMARY OUTPUT Regression Statistics Multiple R 0.992 R Square 0.984 Adjusted R Square 0.979 Standard Error 2.26743 Observations 20 ANOVA df SS MS F Signif F Regression 4 4609.83164 1152.45791 224.160 0.0001 Residual 15 77.11836 5.14122 Total 19 4686.95000 Coeff StdError t Stat p -value Intercept -9.611198 2.77988638 -3.457 0.0035 Age 1.327695 0.11491930 11.553 0.0001 Exper -0.106705 0.14265559 -0.748 0.4660 Degrees 7.311332 0.80324187 9.102 0.0001 Prevjobs -0.504168 0.44771573 -1.126 0.2778 -Referring to Table 14-8, the analyst wants to use a t test to test for the significance of the coefficient of X₃. For a level of significance of 0.01, the critical values of the test are ________.

TABLE 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) X₁ = Length of time in weight-loss program (in months) X₂ = 1 if morning session, 0 if not X₃ = 1 if afternoon session, 0 if not (Base level = evening session) Data for 12 clients on a weight-loss program at the clinic were collected and used to fit the interaction model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₁X₂ + β₅X₁X₂ + ε Partial output from Microsoft Excel follows: Regression Statistics Multiple R 0.73514 R Square 0.540438 Adjusted R Square 0.157469 Standard Error 12.4147 Observations 12 ANOVA $F = 5.41118 \quad$ Significance $F = 0.040201$ Coeff StdError t Stat p -value Intercept 0.089744 14.127 0.0060 0.9951 Length 6.22538 2.43473 2.54956 0.0479 Morn Ses 2.217272 22.1416 0.100141 0.9235 Aft Ses 11.8233 3.1545 3.558901 0.0165 Length*Morn Ses 0.77058 3.562 0.216334 0.8359 Length Aft Ses -0.54147 3.35988 -0.161158 0.8773 -Referring to Table 14-11, in terms of the ?s in the model, give the mean change in weight-loss (Y)for every 1 month increase in time in the program (X?)when attending the morning session.

TABLE 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), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. The coefficients of partial determination ( 2 Yj. (Allvariables except $j$ ) ) of each of the 6 predictors are, respectively, 0.2807, 0.0386, 0.0317, 0.0141, 0.0958, and 0.1201. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error 40 Observations $\text { ANOVA }$ df SS MS F significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 $\text { ANOVA }$ df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 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 Table 14-17 Model 1, which of the following is the correct null hypothesis to test whether being married or not makes a difference in the mean number of weeks a worker is unemployed due to a layoff while holding constant the effect of all the other independent variables?

TABLE 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), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. The coefficients of partial determination ( 2 Yj. (Allvariables except $j$ ) ) of each of the 6 predictors are, respectively, 0.2807, 0.0386, 0.0317, 0.0141, 0.0958, and 0.1201. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error 40 Observations $\text { ANOVA }$ df SS MS F significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 $\text { ANOVA }$ df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 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 Table 14-17 Model 1, which of the following is a correct statement?

TABLE 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), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. The coefficients of partial determination ( 2 Yj. (Allvariables except $j$ ) ) of each of the 6 predictors are, respectively, 0.2807, 0.0386, 0.0317, 0.0141, 0.0958, and 0.1201. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error 40 Observations $\text { ANOVA }$ df SS MS F significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 $\text { ANOVA }$ df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 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 Table 14-17 Model 1, which of the following is a correct statement?

TABLE 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), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂= Salaries and X₃= Spending: Regression Statistics Multiple R 0.7930 R Square 0.6288 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 $\text { ANOVA }$ df SS MS Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 0.0000 -957.3401 -549.5050 \% Attendance 8.5014 1.0771 7.8929 0.0000 6.3292 10.6735 Salary 0.000000685 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 -Referring to Table 14-15, which of the following is the correct alternative hypothesis to test whether instructional spending per pupil has any effect on percentage of students passing the proficiency test, taking into account the effect of all the other independent variables?

TABLE 14-4 A real estate builder wishes to determine how house size (House) is influenced by family income (Income), family size (Size), and education of the head of household (School). House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is in years. The builder randomly selected 50 families and ran the multiple regression. Microsoft Excel output is provided below: SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA df SS MS F Signif F Regression 3605.7736 1201.9245 0.0000 Residual 1214.2264 26.3962 Total 49 4820.0000 Coeff StdError t Stat p -value Intercept -1.6335 5.8078 -0.281 0.7798 Income 0.4485 0.1137 3.9545 0.0003 Size 4.2615 0.8062 5.286 0.0001 School -0.6517 0.4319 -1.509 0.1383 -Referring to Table 14-4, suppose the builder wants to test whether the coefficient on School is significantly different from 0. What is the value of the relevant t-statistic?

TABLE 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) X₁ = Length of time in weight-loss program (in months) X₂ = 1 if morning session, 0 if not X₃ = 1 if afternoon session, 0 if not (Base level = evening session) Data for 12 clients on a weight-loss program at the clinic were collected and used to fit the interaction model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₁X₂ + β₅X₁X₂ + ε Partial output from Microsoft Excel follows: Regression Statistics Multiple R 0.73514 R Square 0.540438 Adjusted R Square 0.157469 Standard Error 12.4147 Observations 12 ANOVA $F = 5.41118 \quad$ Significance $F = 0.040201$ Coeff StdError t Stat p -value Intercept 0.089744 14.127 0.0060 0.9951 Length 6.22538 2.43473 2.54956 0.0479 Morn Ses 2.217272 22.1416 0.100141 0.9235 Aft Ses 11.8233 3.1545 3.558901 0.0165 Length*Morn Ses 0.77058 3.562 0.216334 0.8359 Length Aft Ses -0.54147 3.35988 -0.161158 0.8773 -Referring to Table 14-11, in terms of the ?s in the model, give the mean change in weight-loss (Y)for every 1 month increase in time in the program (X?)when attending the afternoon session.

TABLE 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), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. The coefficients of partial determination ( 2 Yj. (Allvariables except $j$ ) ) of each of the 6 predictors are, respectively, 0.2807, 0.0386, 0.0317, 0.0141, 0.0958, and 0.1201. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error 40 Observations $\text { ANOVA }$ df SS MS F significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 $\text { ANOVA }$ df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 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 Table 14-17 and using both Model 1 and Model 2, there is sufficient evidence to conclude that at least one of the independent variables that are not significant individually has become significant as a group in explaining the variation in the dependent variable at a 5% level of significance?

TABLE 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 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 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 0.8013 R Square 0.6421 Adjusted R Square 0.6313 Standard Error 1.0507 Observations 171 $\text { ANOVA }$ df SS MS F Significance F Regression 5 326.8700 65.3740 59.2168 0.0000 Residual 165 182.1564 1.1040 Total 170 509.0263 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 12.8627 1.0927 11.7713 0.0000 10.7052 15.0202 Cargo Vol 0.0259 0.0102 2.5518 0.0116 0.0059 0.0460 HP -0.0200 0.0018 -11.3307 0.0000 -0.0235 -0.0165 MPG -0.0620 0.0303 -2.0464 0.0423 -0.1218 -0.0022 SUV 0.7679 0.4314 1.7802 0.0769 -0.0838 1.6196 Sedan 0.6427 0.2790 2.3034 0.0225 0.0918 1.1935 The various residual plots are as shown below. The coefficients of partial determination $\left( R ^ { 2 }_{Y j} \right.$ . (All variables except $\left. j \right)$ of each of the 5 predictors are, respectively, $0.0380,0.4376,0.0248,0.0188$ , and $0.0312$ . The coefficient of multiple determination for the regression model using each of the 5 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables $\left( R _ { j } ^ { 2 } \right)$ are, respectively, $0.7461,0.5676,0.6764,0.8582,0.6632$ . -Referring to 14-16, the 0 to 60 miles per hour acceleration time of a sedan is predicted to be 0.6427 seconds higher than that of an SUV.

TABLE 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), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂= Salaries and X₃= Spending: Regression Statistics Multiple R 0.7930 R Square 0.6288 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 $\text { ANOVA }$ df SS MS Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 0.0000 -957.3401 -549.5050 \% Attendance 8.5014 1.0771 7.8929 0.0000 6.3292 10.6735 Salary 0.000000685 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 -Referring to Table 14-15, you can conclude that average teacher salary individually has no impact on the mean percentage of students passing the proficiency test, taking into account the effect of all the other independent variables, at a 10% level of significance based solely on the 95% confidence interval estimate for ??.

TABLE 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), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. The coefficients of partial determination ( 2 Yj. (Allvariables except $j$ ) ) of each of the 6 predictors are, respectively, 0.2807, 0.0386, 0.0317, 0.0141, 0.0958, and 0.1201. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error 40 Observations $\text { ANOVA }$ df SS MS F significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 $\text { ANOVA }$ df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 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 Table 14-17 Model 1, there is sufficient evidence that the number of weeks a worker is unemployed due to a layoff depends on at least one of the explanatory variables at a 10% level of significance.

A regression had the following results: SST = 82.55, SSE = 29.85. It can be said that 63.84% of the variation in the dependent variable is explained by the independent variables in the regression.

TABLE 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 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 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 0.8013 R Square 0.6421 Adjusted R Square 0.6313 Standard Error 1.0507 Observations 171 $\text { ANOVA }$ df SS MS F Significance F Regression 5 326.8700 65.3740 59.2168 0.0000 Residual 165 182.1564 1.1040 Total 170 509.0263 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 12.8627 1.0927 11.7713 0.0000 10.7052 15.0202 Cargo Vol 0.0259 0.0102 2.5518 0.0116 0.0059 0.0460 HP -0.0200 0.0018 -11.3307 0.0000 -0.0235 -0.0165 MPG -0.0620 0.0303 -2.0464 0.0423 -0.1218 -0.0022 SUV 0.7679 0.4314 1.7802 0.0769 -0.0838 1.6196 Sedan 0.6427 0.2790 2.3034 0.0225 0.0918 1.1935 The various residual plots are as shown below. The coefficients of partial determination $\left( R ^ { 2 }_{Y j} \right.$ . (All variables except $\left. j \right)$ of each of the 5 predictors are, respectively, $0.0380,0.4376,0.0248,0.0188$ , and $0.0312$ . The coefficient of multiple determination for the regression model using each of the 5 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables $\left( R _ { j } ^ { 2 } \right)$ are, respectively, $0.7461,0.5676,0.6764,0.8582,0.6632$ . -Referring to 14-16, ________ of the variation in Accel Time can be explained by HP while controlling for the other independent variables.

TABLE 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), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. The coefficients of partial determination ( 2 Yj. (Allvariables except $j$ ) ) of each of the 6 predictors are, respectively, 0.2807, 0.0386, 0.0317, 0.0141, 0.0958, and 0.1201. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error 40 Observations $\text { ANOVA }$ df SS MS F significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 $\text { ANOVA }$ df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 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 Table 14-17 Model 1, which of the following is a correct statement?

TABLE 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 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 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 0.8013 R Square 0.6421 Adjusted R Square 0.6313 Standard Error 1.0507 Observations 171 $\text { ANOVA }$ df SS MS F Significance F Regression 5 326.8700 65.3740 59.2168 0.0000 Residual 165 182.1564 1.1040 Total 170 509.0263 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 12.8627 1.0927 11.7713 0.0000 10.7052 15.0202 Cargo Vol 0.0259 0.0102 2.5518 0.0116 0.0059 0.0460 HP -0.0200 0.0018 -11.3307 0.0000 -0.0235 -0.0165 MPG -0.0620 0.0303 -2.0464 0.0423 -0.1218 -0.0022 SUV 0.7679 0.4314 1.7802 0.0769 -0.0838 1.6196 Sedan 0.6427 0.2790 2.3034 0.0225 0.0918 1.1935 The various residual plots are as shown below. The coefficients of partial determination $\left( R ^ { 2 }_{Y j} \right.$ . (All variables except $\left. j \right)$ of each of the 5 predictors are, respectively, $0.0380,0.4376,0.0248,0.0188$ , and $0.0312$ . The coefficient of multiple determination for the regression model using each of the 5 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables $\left( R _ { j } ^ { 2 } \right)$ are, respectively, $0.7461,0.5676,0.6764,0.8582,0.6632$ . -Referring to 14-16, which of the following assumptions is most likely violated based on the residual plot for HP?