Exam 15: Multiple Regression Model Building

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,the residual plot suggests that a nonlinear model on % attendance may be a better model.

The logarithm transformation can be used

Two simple regression models were used to predict a single dependent variable.Both models were highly significant,but when the two independent variables were placed in the same multiple regression model for the dependent variable,R² did not increase substantially and the parameter estimates for the model were not significantly different from 0.This is probably an example of collinearity.

TABLE 15-2 In Hawaii,condemnation proceedings are under way to enable private citizens to own the property that their homes are built on.Until recently,only estates were permitted to own land,and homeowners leased the land from the estate.In order to comply with the new law,a large Hawaiian estate wants to use regression analysis to estimate the fair market value of the land.The following model was fit to data collected for n = 20 properties,10 of which are located near a cove. -Referring to Table 15-2,is the overall model statistically adequate at a 0.05 level of significance for predicting sale price (Y)?

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,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.

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,X₃,X₅ and X₆?

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,the prediction of time to relief for a person receiving a dose of the drug 10 units above the mean dose (i.e.,the prediction of Y for X = 10)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,the model that includes X₁,X₂,X₅ and X₆ should be among the appropriate models using the Mallow's C_p statistic.

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,does there appear to be significant upward curvature in the response curve relating the demand (Y)and the price (X)at 10% level of significance?

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 an F test to determine if there is a significant curvilinear relationship between time and dose.If she chooses to use a level of significance of 0.05,she would decide that there is a significant curvilinear relationship.

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,the better model using a 5% level of significance derived from the "best" model above 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,the variable X₆ should be dropped to remove collinearity.

The ________ (larger/smaller)the value of the Variance Inflationary Factor,the higher is the collinearity of the X variables.

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,X₃,X₅ and X₆ should be among the appropriate models using the Mallow's C_p 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,the "best" model chosen using the adjusted R-square statistic is

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,a more parsimonious simple linear model is likely to be statistically superior to the fitted curvilinear for predicting sale price (Y).

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,there is reason to suspect collinearity between some pairs of predictors based on the values of the variance inflationary factor.

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 among the appropriate models using the Mallow's C_p statistic.

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 SUV?

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,X₂,X₅ and X₆?