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Exam 14: Introduction to Multiple Regression

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Question 1

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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),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,X₁ = % Attendance,X₂= Salaries and X₃= Spending: -Referring to Table 14-15,you can conclude that instructional spending per pupil individually 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 1% level of significance based solely on the 95% confidence interval estimate for β₃.

A) True
B) False

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Question 2

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TABLE 14-4 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 14-4,one individual in the sample had an annual income of $100,000,a family size of 10,and an education of 16 years.This individual owned a home with an area of 7,000 square feet (House = 70.00) .What is the residual (in hundreds of square feet) for this data point?

A) 7.40
B) 2.52
C) -2.52
D) -5.40

E) A) and B)
F) None of the above

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Question 3

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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 X₃ = 1 if afternoon session,0 if not (Base level = evening session) Data for 12 clients on a weight-loss program at the clinic were collected and used to fit the interaction model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₁X₂ + β₅X₁X₂ + ε Partial 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 in the program (X₁) when attending the afternoon session.

TABLE 14-9 You decide to predict gasoline prices in different cities and towns in the United States for your term project.Your dependent variable is price of gasoline per gallon and your explanatory variables are per capita income,the number of firms that manufacture automobile parts in and around the city,the number of new business starts in the last year,population density of the city,percentage of local taxes on gasoline,and the number of people using public transportation.You collected data of 32 cities and obtained a regression sum of squares SSR = 122.8821.Your computed value of standard error of the estimate is 1.9549. -Referring to Table 14-9,if variables that measure the number of new business starts in the last year and population density of the city were removed from the multiple regression model,which of the following would be true?

TABLE 14-10 You worked as an intern at We Always Win Car Insurance Company last summer.You notice 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 Excel and obtained the following information: -Referring to Table 14-10,the 99% confidence interval for the change in mean insurance premiums of a person who has become 1 year older (i.e.,the slope coefficient for AGE)is ?-0.82 ± ________.

TABLE 14-17 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.The coefficients of partial determination (R )of each of the 6 predictors are,respectively,0.2807,0.0386,0.0317,0.0141,0.0958,and 0.1201. 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 14-17 Model 1,predict the number of weeks being unemployed due to a layoff for a worker who is a thirty-year old,has 10 years of education,has 15 years of experience at the previous job,is married,is the head of household and is a manager.

TABLE 14-10 You worked as an intern at We Always Win Car Insurance Company last summer.You notice 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 Excel and obtained the following information: -Referring to Table 14-10,to test the significance of the multiple regression model,the null hypothesis should be rejected while allowing for 1% probability of committing a type I error.

TABLE 14-4 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 14-4,what is the value of the calculated F test statistic that is missing from the output for testing whether the whole regression model is significant?

TABLE 14-4 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 14-4,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 14-17 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.The coefficients of partial determination (R )of each of the 6 predictors are,respectively,0.2807,0.0386,0.0317,0.0141,0.0958,and 0.1201. 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 14-17 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 14-13 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 14-13,the predicted demand in San Francisco when the mortgage rate is 10% is ________.

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),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,X₁ = % Attendance,X₂= Salaries and X₃= Spending: -Referring to Table 14-15,what is the p-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 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),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,X₁ = % Attendance,X₂= Salaries and X₃= Spending: -Referring to Table 14-15,the null hypothesis H₀: = β₁ = β₂ = β₃ = 0 implies that percentage of students passing the proficiency test is not related to any of the explanatory variables.

TABLE 14-17 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.The coefficients of partial determination (R ) of each of the 6 predictors are,respectively,0.2807,0.0386,0.0317,0.0141,0.0958,and 0.1201. 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 14-17 Model 1,which of the following is a correct statement?

When an additional explanatory variable is introduced into a multiple regression model,the coefficient of multiple determination will never decrease.

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 14-19 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; 15 did not have a lawn service (code 0)and 15 had a lawn service (code 1).Additional information available concerning these 30 home owners includes family income (Income,in thousands of dollars),lawn size (Lawn Size,in thousands of square feet),attitude toward outdoor recreational activities (Atitude 0 = unfavorable,1 = favorable),number of teenagers in the household (Teenager),and age of the head of the household (Age). The Minitab 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 Age makes a significant contribution to the model in the presence of the other independent variables at a 0.05 level of significance?

In calculating the standard error of the estimate,SYX = ,there are n - k - 1 degrees of freedom,where n is the sample size and k represents the number of independent variables in the model.

TABLE 14-17 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.The coefficients of partial determination (R )of each of the 6 predictors are,respectively,0.2807,0.0386,0.0317,0.0141,0.0958,and 0.1201. 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 14-17 Model 1,the null hypothesis H₀: β₁ = β₂ = β₃ = β₄ = β₅ = β₆ = 0 implies that the number of weeks a worker is unemployed due to a layoff is not related to any of the explanatory variables.

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 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 coefficients of partial determination (R )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 Xⱼ as the dependent variable and all other X variables as independent variables (R )are,respectively,0.7461,0.5676,0.6764,0.8582,0.6632. -Referring to 14-16,________ of the variation in Accel Time can be explained by HP while controlling for the other independent variables.