Exam 13: Introduction to Multiple Regression

TABLE 13-16 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. The various residual plots are as shown below. The coefficient of multiple determination for the regression model using each of the 5 variables Xj as the dependent variable and all other X variables as independent variables (Rj2) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 13-16, what is the p-value of the test statistic to determine whether HP makes a significant contribution to the regression model in the presence of the other independent variables at a 5% level of significance?

TABLE 13-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and four 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: -Referring to Table 13-8, the value of the coefficient of multiple determination, r²Y.₁₂₃₄, is ________.

TABLE 13-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and four 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: -Referring to Table 13-8, the F test for the significance of the entire regression performed at a level of significance of 0.01 leads to a rejection of the null hypothesis.

TABLE 13-10 You worked as an intern at We Always Win Car Insurance Company last summer. You noticed that individual car insurance premiums depend very much on the age of the individual, the number of traffic tickets received by the individual, and the population density of the city in which the individual lives. You performed a regression analysis in Microsoft Excel and obtained the following information: -Referring to Table 13-10, the multiple regression model is significant at a 10% level of significance.

TABLE 13-17 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. 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: -Referring to Table 13-17 Model 1, we can conclude that, holding constant the effect of the other independent variables, there is a difference in the mean number of weeks a worker is unemployed due to a layoff between a worker who is married and one who is not at a 1% level of significance if all we have is the information of the 95% confidence interval estimate for β₄.

TABLE 13-4 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: -Referring to Table 13-4, what minimum annual income would an individual with a family size of 4 and 16 years of education need to attain a predicted 10,000 square foot home (House = 100)?

TABLE 13-17 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. 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: -Referring to Table 13-17 Model 1, what is the standard error of estimate?

TABLE 13-16 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. The various residual plots are as shown below. The coefficient of multiple determination for the regression model using each of the 5 variables Xj as the dependent variable and all other X variables as independent variables (Rj2) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 13-16, what is the value of the test statistic to determine whether 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 13-15 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, = : % Attendance, = Salaries and = Spending: -Referring to Table 13-15, 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?

TABLE 13-15 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, = : % Attendance, = Salaries and = Spending: -Referring to Table 13-15, there is sufficient evidence that the percentage of students passing the proficiency test depends on all of the explanatory variables at a 5% level of significance.

TABLE 13-16 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. The various residual plots are as shown below. The coefficient of multiple determination for the regression model using each of the 5 variables Xj as the dependent variable and all other X variables as independent variables (Rj2) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 13-16, there is enough evidence to conclude that HP makes a significant contribution to the regression model in the presence of the other independent variables at a 5% level of significance.

TABLE 13-16 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. The various residual plots are as shown below. The coefficient of multiple determination for the regression model using each of the 5 variables Xj as the dependent variable and all other X variables as independent variables (Rj2) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 13-16, the 0 to 60 miles per hour acceleration time of a coupe is predicted to be 0.7679 seconds higher than that of a sedan.

TABLE 13-16 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. The various residual plots are as shown below. The coefficient of multiple determination for the regression model using each of the 5 variables Xj as the dependent variable and all other X variables as independent variables (Rj2) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 13-16, what is the p-value of the test statistic to determine whether 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 13-10 You worked as an intern at We Always Win Car Insurance Company last summer. You noticed that individual car insurance premiums depend very much on the age of the individual, the number of traffic tickets received by the individual, and the population density of the city in which the individual lives. You performed a regression analysis in Microsoft Excel and obtained the following information: -Referring to Table 13-10, to test the significance of the multiple regression model, the p-value of the test statistic in the sample is ________.

TABLE 13-12 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 one of the three jurisdictions: on campus, in downtown and off campus, or outside of downtown and off campus. The population regression model hypothesized is Yi = α + β1X1i + β2X2i + β3X3i + ε where Y is the meter price X1 is the number of blocks to the quad X2 is a dummy variable that takes the value 1 if the meter is located in downtown and off campus and the value 0 otherwise X3 is a dummy variable that takes the value 1 if the meter is located outside of downtown and off campus, and the value 0 otherwise The following Microsoft Excel results are obtained. -Referring to Table 13-12, if one is already outside of downtown and off campus but decides to park three more blocks from the quad, the estimated mean parking meter rate will

TABLE 13-15 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, = : % Attendance, = Salaries and = Spending: -Referring to Table 13-15, you can conclude that instructional spending per pupil 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 5% level of significance using the 95% confidence interval estimate for β₃.

TABLE 13-13 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 + 5X1 + 8X2 where X1 = mortgage rate in % X2 = 1 if SF, 0 if LA Y = demand in $100 per capita -Referring to Table 13-13, the effect of living in San Francisco rather than Los Angeles is to increase the mean demand by an estimated ________ per capita.

TABLE 13-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and four 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: -Referring to Table 13-8, the estimate of the unit change in the mean of Y per unit change in X₄, taking into account the effects of the other three variables, is ________.

TABLE 13-7 The department head of the accounting department wanted to see if she could predict the GPA of students using the number of course units (credits) and total SAT scores of each. She takes a sample of students and generates the following Microsoft Excel output: -Referring to Table 13-7, the department head wants to use a t test to test for the significance of the coefficient of X₁. For a level of significance of 0.05, the critical values of the test are ________.

TABLE 13-16 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. The various residual plots are as shown below. The coefficient of multiple determination for the regression model using each of the 5 variables Xj as the dependent variable and all other X variables as independent variables (Rj2) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 13-16, what is the correct interpretation for the estimated coefficient for HP?