Exam 14: Introduction to Multiple Regression

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 Data for 25 clients on a weight-loss program at the clinic were collected and used to fit the interaction model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₁X₂ + ε Output from Microsoft Excel follows: -Referring to Table 14-11,in terms of the βs in the model,give the mean change in weight loss (Y)for every 1 month increase in time on the program (X₁)when attending the morning session.

TABLE 14-4 A real estate builder wishes to determine how house size (House)is influenced by family income (Income)and family size (Size).House size is measured in hundreds of square feet and income is measured in thousands of dollars.The builder randomly selected 50 families and ran the multiple regression.Partial Microsoft Excel output is provided below: Also SSR (X₁ ∣ X₂)= 36400.6326 and SSR (X₂ ∣ X₁)= 3297.7917 -Referring to Table 14-4,at the 0.01 level of significance,what conclusion should the builder draw regarding the inclusion of Size in the regression model?

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; 11 did not have a lawn service (code 0)and 19 had a lawn service (code 1).Additional information available concerning these 30 home owners includes family income (Income,in thousands of dollars)and lawn size (Lawn Size,in thousands of square feet). The PHStat output is given below: -Referring to Table 14-19,what is the estimated probability that a home owner with a family income of $100,000 and a lawn size of 2,000 square feet will purchase a lawn service?

TABLE 14-8 A financial analyst wanted to examine the relationship between salary (in $1,000)and 2 variables: age (X₁ = Age)and experience in the field (X₂ = Exper).He took a sample of 20 employees and obtained the following Microsoft Excel output: Also,the sum of squares due to the regression for the model that includes only Age is 5022.0654 while the sum of squares due to the regression for the model that includes only Exper is 125.9848. -Referring to Table 14-8,the partial F test for H₀ : Variable X₂ does not significantly improve the model after variable X₁ has been included H₁ Variable X₂ significantly improves the model after variable X₁ has been included has ________ and ________ degrees of freedom.

TABLE 14-7 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 14-7,the department head wants to test H₀ : β₁ = β₂ = 0.The appropriate alternative hypothesis is ________.

TABLE 14-4 A real estate builder wishes to determine how house size (House)is influenced by family income (Income)and family size (Size).House size is measured in hundreds of square feet and income is measured in thousands of dollars.The builder randomly selected 50 families and ran the multiple regression.Partial Microsoft Excel output is provided below: Also SSR (X₁ ∣ X₂)= 36400.6326 and SSR (X₂ ∣ X₁)= 3297.7917 -Referring to Table 14-4,what are the residual degrees of freedom that are missing from the output?

TABLE 14-2 A professor of industrial relations believes that an individual's wage rate at a factory (Y)depends on his performance rating (X₁)and the number of economics courses the employee successfully completed in college (X₂).The professor randomly selects 6 workers and collects the following information: -Referring to Table 14-2,for these data,what is the value for the regression constant,b₀?

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 30 different vehicle models were collected: Y (Accel Time): Acceleration time in sec. X₁ (Engine Size): c.c. X₂ (Sedan): 1 if the vehicle model is a sedan and 0 otherwise 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 partial determinations and are 0.3301,and 0.0594,respectively. The coefficient of determination for the regression model using each of the 2 independent variables as the dependent variable and the other independent variable as independent variables ( )are,respectively 0.0077,and 0.0077. -Referring to Table 14-16,what is the value of the test statistic to determine whether engine size makes a significant contribution to the regression model in the presence of the other independent variable at a 5% level of significance?

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),mean teacher salary in thousands of dollars (Salaries),and instructional spending per pupil in thousands of dollars (Spending)of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable,X₁ = Salaries and X₂ = Spending: -True or False: Referring to Table 14-15,you can conclude that mean teacher salary has no impact on the mean percentage of students passing the proficiency test,taking into account the effect of instructional spending per pupil,at a 5% level of significance using the confidence interval estimate for β₁.

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; 11 did not have a lawn service (code 0)and 19 had a lawn service (code 1).Additional information available concerning these 30 home owners includes family income (Income,in thousands of dollars)and lawn size (Lawn Size,in thousands of square feet). The PHStat output is given below: -Referring to Table 14-19,what should be the decision ('reject' or 'do not reject')on the null hypothesis when testing whether Income makes a significant contribution to the model in the presence of LawnSize 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 30 different vehicle models were collected: Y (Accel Time): Acceleration time in sec. X₁ (Engine Size): c.c. X₂ (Sedan): 1 if the vehicle model is a sedan and 0 otherwise 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 partial determinations and are 0.3301,and 0.0594,respectively. The coefficient of determination for the regression model using each of the 2 independent variables as the dependent variable and the other independent variable as independent variables ( )are,respectively 0.0077,and 0.0077. -Referring to Table 14-16,what is the correct interpretation for the estimated coefficient for X₂?

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),mean teacher salary in thousands of dollars (Salaries),and instructional spending per pupil in thousands of dollars (Spending)of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable,X₁ = Salaries and X₂ = Spending: -True or False: Referring to Table 14-15,the null hypothesis H₀ : β₁ = β₂ = 0 implies that percentage of students passing the proficiency test is not related to one of the explanatory variables.

TABLE 14-18 A logistic regression model was estimated in order to predict the probability that a randomly chosen university or college would be a private university using information on mean total Scholastic Aptitude Test score (SAT)at the university or college and whether the TOEFL criterion is at least 90 (Toefl90 = 1 if yes,0 otherwise).The dependent variable,Y,is school type (Type = 1 if private and 0 otherwise). The PHStat output is given below: -True or False: Referring to Table 14-18,there is not enough evidence to conclude that Toefl90 makes a significant contribution to the model in the presence of SAT at a 0.05 level of significance.

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; 11 did not have a lawn service (code 0)and 19 had a lawn service (code 1).Additional information available concerning these 30 home owners includes family income (Income,in thousands of dollars)and lawn size (Lawn Size,in thousands of square feet). The PHStat output is given below: -True or False: 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 Income at a 0.05 level of significance.

TABLE 14-3 An economist is interested to see how consumption for an economy (in $ billions)is influenced by gross domestic product ($ billions)and aggregate price (consumer price index).The Microsoft Excel output of this regression is partially reproduced below. -Referring to Table 14-3,to test for the significance of the coefficient on gross domestic product,the p-value is

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 30 different vehicle models were collected: Y (Accel Time): Acceleration time in sec. X₁ (Engine Size): c.c. X₂ (Sedan): 1 if the vehicle model is a sedan and 0 otherwise 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 partial determinations and are 0.3301,and 0.0594,respectively. The coefficient of determination for the regression model using each of the 2 independent variables as the dependent variable and the other independent variable as independent variables ( )are,respectively 0.0077,and 0.0077. -Referring to Table 14-16,which of the following assumptions is most likely violated based on the residual plot for Engine Size?

TABLE 14-5 A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies.She proceeds to randomly select 26 large corporations and record information in millions of dollars.The Microsoft Excel output below shows results of this multiple regression. -Referring to Table 14-5,suppose the microeconomist wants to test whether the coefficient on Capital is significantly different from 0.What is the value of the relevant t-statistic?

TABLE 14-8 A financial analyst wanted to examine the relationship between salary (in $1,000)and 2 variables: age (X₁ = Age)and experience in the field (X₂ = Exper).He took a sample of 20 employees and obtained the following Microsoft Excel output: Also,the sum of squares due to the regression for the model that includes only Age is 5022.0654 while the sum of squares due to the regression for the model that includes only Exper is 125.9848. -Referring to Table 14-8,________% of the variation in salary can be explained by the variation in experience while holding age constant.

TABLE 14-18 A logistic regression model was estimated in order to predict the probability that a randomly chosen university or college would be a private university using information on mean total Scholastic Aptitude Test score (SAT)at the university or college and whether the TOEFL criterion is at least 90 (Toefl90 = 1 if yes,0 otherwise).The dependent variable,Y,is school type (Type = 1 if private and 0 otherwise). The PHStat output is given below: -Referring to Table 14-18,what should be the decision ('reject' or 'do not reject')on the null hypothesis when testing whether Toefl90 makes a significant contribution to the model in the presence of SAT at a 0.05 level of significance?

TABLE 14-6 One of the most common questions of prospective house buyers pertains to the cost of heating in dollars (Y).To provide its customers with information on that matter,a large real estate firm used the following 2 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit (X₁)and the amount of insulation in inches (X₂).Given below is EXCEL output of the regression model. Also SSR (X₁ ∣ X₂)= 8343.3572 and SSR (X₂ ∣ X₁)= 4199.2672 -Referring to Table 14-6 and allowing for a 1% probability of committing a type I error,what is the decision and conclusion for the test H₀ : β₁ = β₂ = 0 vs.H₁ : At least one β_j ≠ 0,j = 1,2?