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 coefficient of multiple determination (R )for the regression model using each of the 5 variables Xⱼ 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 variance inflationary factor of Sedan?

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 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. Let Y = % Passing as the dependent variable,X₁ = % Attendance,X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination (R )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 15-4 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 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. Let Y = % Passing as the dependent variable,X₁ = % Attendance,X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination (R )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,what are,respectively,the values of the variance inflationary factor of the 3 predictors?

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 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. Let Y = % Passing as the dependent variable,X₁ = % Attendance,X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination (R )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,what is the value of the test statistic to determine whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance?

The parameter estimates are biased when collinearity is present in a multiple regression equation.

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: Y = β₀ + β₁X + β₁X² + ε 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: -Referring to Table 15-1,what is the p-value associated with the test statistic for testing whether there is an upward curvature in the response curve relating the demand (Y)and the price (X)?

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: Y = β₀ + β₁X + β₁X² + ε 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: -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)?

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y)and the independent variables are the age of the worker (X₁),the number of years of education received (X₂),the number of years at the previous job (X₃),a dummy variable for marital status (X₄: 1 = married,0 = otherwise),a dummy variable for head of household (X₅: 1 = yes,0 = no)and a dummy variable for management position (X₆: 1 = yes,0 = no). The coefficient of multiple determination (R²_J )for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are,respectively,0.2628,0.1240,0.2404,0.3510,0.3342 and 0.0993. The partial results from best-subset regression are given below: -Referring to Table 15-6,the model that includes X₁,X₂,X₃,X₅ and X₆ should be selected using the adjusted r² statistic.

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 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. Let Y = % Passing as the dependent variable,X₁ = % Attendance,X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination (R )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,what is the p-value of the test statistic to determine whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance?

Which of the following is used to find a "best" model?

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y)and the independent variables are the age of the worker (X₁),the number of years of education received (X₂),the number of years at the previous job (X₃),a dummy variable for marital status (X₄: 1 = married,0 = otherwise),a dummy variable for head of household (X₅: 1 = yes,0 = no)and a dummy variable for management position (X₆: 1 = yes,0 = no). The coefficient of multiple determination (R²_J )for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are,respectively,0.2628,0.1240,0.2404,0.3510,0.3342 and 0.0993. The partial results from best-subset regression are given below: -Referring to Table 15-6,what is the value of the Mallow's C_p statistic for the model that includes all the six independent variables?

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

Collinearity is present if the dependent variable is linearly related to one of the explanatory variables.

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." -Referring to Table 15-3,suppose the chemist decides to use a t test to determine if the linear term is significant.The value of the test statistic is ________.

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y)and the independent variables are the age of the worker (X₁),the number of years of education received (X₂),the number of years at the previous job (X₃),a dummy variable for marital status (X₄: 1 = married,0 = otherwise),a dummy variable for head of household (X₅: 1 = yes,0 = no)and a dummy variable for management position (X₆: 1 = yes,0 = no). The coefficient of multiple determination (R²_J )for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are,respectively,0.2628,0.1240,0.2404,0.3510,0.3342 and 0.0993. The partial results from best-subset regression are given below: -Referring to Table 15-6,what is the value of the variance inflationary factor of Edu?

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." -Referring to Table 15-3,suppose the chemist decides to use a t test to determine if the linear term is significant.Using a level of significance of 0.05,she would decide that the linear term is significant.

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y)and the independent variables are the age of the worker (X₁),the number of years of education received (X₂),the number of years at the previous job (X₃),a dummy variable for marital status (X₄: 1 = married,0 = otherwise),a dummy variable for head of household (X₅: 1 = yes,0 = no)and a dummy variable for management position (X₆: 1 = yes,0 = no). The coefficient of multiple determination (R²_J )for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are,respectively,0.2628,0.1240,0.2404,0.3510,0.3342 and 0.0993. The partial results from best-subset regression are given below: -Referring to Table 15-6,the variable X₄ should be dropped to remove collinearity.

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y)and the independent variables are the age of the worker (X₁),the number of years of education received (X₂),the number of years at the previous job (X₃),a dummy variable for marital status (X₄: 1 = married,0 = otherwise),a dummy variable for head of household (X₅: 1 = yes,0 = no)and a dummy variable for management position (X₆: 1 = yes,0 = no). The coefficient of multiple determination (R²_J )for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are,respectively,0.2628,0.1240,0.2404,0.3510,0.3342 and 0.0993. The partial results from best-subset regression are given below: -Referring to Table 15-6,what is the value of the variance inflationary factor of Job Yr?

If 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 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y)and the independent variables are the age of the worker (X₁),the number of years of education received (X₂),the number of years at the previous job (X₃),a dummy variable for marital status (X₄: 1 = married,0 = otherwise),a dummy variable for head of household (X₅: 1 = yes,0 = no)and a dummy variable for management position (X₆: 1 = yes,0 = no). The coefficient of multiple determination (R²_J )for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are,respectively,0.2628,0.1240,0.2404,0.3510,0.3342 and 0.0993. The partial results from best-subset regression are given below: -Referring to Table 15-6,what is the value of the Mallow's C_p statistic for the model that includes X₁,X₅ and X₆?