Exam 14: Introduction to Multiple Regression

A regression had the following results: SST = 102.55, SSE = 82.04. It can be said that 90.0% of the variation in the dependent variable is explained by the independent variables in the 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), 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 affected by any of the explanatory variables.

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 14-17 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 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 affected by 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. -Referring to 14-16, the 0 to 60 miles per hour acceleration time of a coupe is predicted to be 0.7679 seconds higher than that of a sedan.

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, there is not enough evidence to conclude that Age makes a significant contribution to the model in the presence of the other independent variables at a 0.05 level of significance.

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 - 0.1X₁X₂ where X₁ = horsepower X₂ = 1 if 4 cylinders, 0 if 6 cylinders Y = mileage. -Referring to Table 14-14, the fitted model for predicting mileages for 6-cylinder cars 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 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. -Referring to 14-16, the error appears to be right-skewed.

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, one economy in the sample had an aggregate consumption level of $4 billion, a GDP of $6 billion, and an aggregate price level of 200. What is the residual for this data point?

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 should be rejected at a 5% level of significance when testing whether daily mean of the percentage of students attending class has any effect on percentage of students passing the proficiency test, taking into account the effect of all the other independent variables.

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, what is the predicted consumption level for an economy with GDP equal to $4 billion and an aggregate price index of 150?

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 p-value of the test is ________.

TABLE 14-17 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 14-5 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, what is the p-value for testing whether Wages have a negative impact on corporate sales?

TABLE 14-17 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, there is sufficient evidence that the number of weeks a worker is unemployed due to a layoff depends on all of the explanatory variables at a 10% 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, what is the estimated mean consumption level for an economy with GDP equal to $2 billion and an aggregate price index of 90?

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.

An interaction term in a multiple regression model may be used when

TABLE 14-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and 4 variables: age (X₁ = Age), experience in the field (X₂ = Exper), number of degrees (X₃ = Degrees), and number of previous jobs in the field (X₄ = Prevjobs). He took a sample of 20 employees and obtained the following Microsoft Excel output: -Referring to Table 14-8, the F test for the significance of the entire regression performed at a level of significance of 0.01 leads to a rejection of the null hypothesis.

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, which of the independent variables in the model are significant at the 5% level?