Exam 15: Multiple Regression Model Building

TABLE 15-5 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. ANOVA The various residual plots are as shown below. The coefficient of partial determination ( ) 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 as the dependent variable and all other X variables as independent variables ( ) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, what are, respectively, the values of the variance inflationary factor of the 5 predictors?

TABLE 15-5 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. ANOVA The various residual plots are as shown below. The coefficient of partial determination ( ) 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 as the dependent variable and all other X variables as independent variables ( ) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, 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 15-4 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 average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Let Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination (R_j ²) of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743. The output from the best-subset regressions is given below: Following is the residual plot for % Attendance: Following is the output of several multiple regression models: Model (I): Model (II): Model (III): -Referring to Table 15-4, the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is not significant at a 5% level of significance.

Which of the following will NOT change a nonlinear model into a linear model?

TABLE 15-1 A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand increases and it decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status owners believe they gain in obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model: where Y = demand (in thousands) and X = retail price per carat. This model was fit to data collected for a sample of 12 rare gems of this type. A portion of the computer analysis obtained from Microsoft Excel is shown below: SUMMARY OUTPUT Regression Statistics ANOVA -Referring to Table 15-1, what is the value of the test statistic for testing whether there is an upward curvature in the response curve relating the demand (Y) and the price (X)?

In multiple regression, the ________ procedure permits variables to enter and leave the model at different stages of its development.

TABLE 15-4 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 average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Let Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination (R_j ²) of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743. The output from the best-subset regressions is given below: Following is the residual plot for % Attendance: Following is the output of several multiple regression models: Model (I): Model (II): Model (III): -Referring to Table 15-4, which of the following predictors should first be dropped to remove collinearity?

TABLE 15-5 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. ANOVA The various residual plots are as shown below. The coefficient of partial determination ( ) 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 as the dependent variable and all other X variables as independent variables ( ) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, 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 15-5 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. ANOVA The various residual plots are as shown below. The coefficient of partial determination ( ) 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 as the dependent variable and all other X variables as independent variables ( ) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, which of the following assumptions is most likely violated based on the normal probability plot?

TABLE 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered." SUMMARY OUTPUT Regression Statistics ANOVA -Referring to Table 15-3, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. The value of the test statistic is ________.

In data mining where huge data sets are being explored to discover relationships among a large number of variables, the best-subsets approach is more practical than the stepwise regression approach.

TABLE 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered." SUMMARY OUTPUT Regression Statistics ANOVA -Referring to Table 15-3, the prediction of time to relief for a person receiving a dose of the drug 10 units above the average dose , is ________.

TABLE 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered." SUMMARY OUTPUT Regression Statistics ANOVA -Referring to Table 15-3, suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a curvilinear model that includes a linear term. If she used a level of significance of 0.05, she would decide that the linear model is sufficient.

TABLE 15-5 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. ANOVA The various residual plots are as shown below. The coefficient of partial determination ( ) 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 as the dependent variable and all other X variables as independent variables ( ) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, 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?

In multiple regression, the ________ procedure permits variables to enter and leave the model at different stages of its development.

TABLE 15-5 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. ANOVA The various residual plots are as shown below. The coefficient of partial determination ( ) 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 as the dependent variable and all other X variables as independent variables ( ) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, what is the p-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 15-5 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. ANOVA The various residual plots are as shown below. The coefficient of partial determination ( ) 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 as the dependent variable and all other X variables as independent variables ( ) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, what is the correct interpretation for the estimated coefficient for HP?

If a group of independent variables are not significant individually but are significant as a group at a specified level of significance, this is most likely due to

TABLE 15-5 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. ANOVA The various residual plots are as shown below. The coefficient of partial determination ( ) 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 as the dependent variable and all other X variables as independent variables ( ) are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, what is the correct interpretation for the estimated coefficient for Sedan?

The goals of model building are to find a good model with the fewest independent variables that is easier to interpret and has lower probability of collinearity.