Exam 14: Introduction to Multiple Regression

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,the p-value for the regression model as a whole 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),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 alternative hypothesis H₁ : At least one of β_j ≠ 0 for j = 1,2 implies that percentage of students passing the proficiency test is affected by both of the explanatory variables.

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 -True or False: When an explanatory variable is dropped from a multiple regression model,the adjusted r² can increase.

TABLE 14-12 As a project for his business statistics class,a student examined the factors that determined parking meter rates throughout the campus area.Data were collected for the price ($)per hour of parking,blocks to the quadrangle,and whether the parking is on or off campus.The population regression model hypothesized is Y_i = α + β₁X_1i + β₂X_2i + ε where Y is the meter price per hour. X₁ is the number of blocks to the quad. X₂ is a dummy variable that takes the value 1 if the meter is located on campus and 0 otherwise. The following Excel results are obtained. -True or False: An interaction term in a multiple regression model may be used when the relationship between X₁ and Y changes for differing values of X₂.

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,when the economist used a simple linear regression model with consumption as the dependent variable and GDP as the independent variable,he obtained an r² value of 0.971.What additional percentage of the total variation of consumption has been explained by including aggregate prices in the multiple regression?

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: -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 14-7 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 use a t test to test for the significance of the coefficient of X₁.The value of the test statistic is ________.

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 analyst wants to use a t test to test for the significance of the coefficient of X₂.The value of the test statistic is ________.

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 whether aggregate price index has a negative impact on consumption,the p-value 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),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: -Referring to Table 14-15,what is the value of the test statistic when testing whether mean teacher salary has any effect on percentage of students passing the proficiency test,taking into account the effect of instructional spending per pupil?

TABLE 14-14 An automotive engineer would like to be able to predict automobile mileages.She believes that the two most important characteristics that affect mileage are horsepower and the number of cylinders (4 or 6)of a car.She believes that the appropriate model is Y = 40 - 0.05X₁ + 20X₂ - 0.1X₁ X₂ where X₁ = horsepower X₂ = 1 if 4 cylinders,0 if 6 cylinders Y = mileage -Referring to Table 14-14,the predicted mileage for a 300 horsepower,6-cylinder car 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,which of the following values for the level of significance is the smallest for which the regression model as a whole is significant?

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)and a dummy variable for management position (Manager: 1 = yes,0 = no). The results of the regression analysis are given below: -Referring to Table 14-17,predict the number of weeks being unemployed due to a layoff for a worker who is a thirty-year-old and is a manager.

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,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 14-8 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 estimated change in the mean salary (in $1,000)when an employee is a year older holding experience constant is ________.

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 and the number of firms that manufacture automobile parts in and around the city.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,what is the value of the coefficient of multiple determination?

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. -True or False: Referring to Table 14-16,there is enough evidence to conclude that 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 14-6 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,________% of the variation in heating cost can be explained by the variation in the amount of insulation while holding the minimum outside temperature constant.

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,which of the following statements is supported by the analysis shown?

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)and a dummy variable for management position (Manager: 1 = yes,0 = no). The results of the regression analysis are given below: -True or False: Referring to Table 14-17,the alternative hypothesis H₁ : At least one of β_j ≠ 0 for j = 1,2 implies that the number of weeks a worker is unemployed due to a layoff is related to at least one of the explanatory variables.