Exam 14: Introduction to Multiple Regression

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: SUMMARY OUTPUT Regression Statistics ANOVA -Referring to Table 14-4, what fraction of the variability in house size is explained by income, size of family, and education?

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 average of the percentage of students attending class (% Attendance), average 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: Note: -Referring to Table 14-15, predict the percentage of students passing the proficiency test for all the schools that have a daily average of 95% of students attending class, an average teacher salary of 40,000 dollars, and an instructional spending per pupil of 2000 dollars.

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 average of the percentage of students attending class (% Attendance), average 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: Note: -Referring to Table 14-15, what is the 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 14-6 One of the most common questions of prospective house buyers pertains to the average cost of heating in dollars (Y). To provide its customers with information on that matter, a large real estate firm used the following 4 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit (X₁), the amount of insulation in inches (X₂), the number of windows in the house (X₃), and the age of the furnace in years (X₄). Given below are the EXCEL outputs of two regression models. Model 1 Note: 2.96869E-05 = 2.96869×10^-5 Model 2 Note: 2.9036E-06 = 2.9036×10^-6 -Referring to Table 14-6, what are the degrees of freedom of the partial F test for H₀ : β₃ = β₄ = 0 vs. H₁ : At least one β_j ≠ 0, j = 3, 4?

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: SUMMARY OUTPUT Regression Statistics ANOVA -Referring to Table 14-8, the analyst wants to use an F-test to test H₀ : β₁ = β₂ = β₃ = β₄ = 0. The appropriate alternative hypothesis 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 average of the percentage of students attending class (% Attendance), average 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: Note: -Referring to Table 14-15, there is sufficient evidence that at least one of the explanatory variables is related to the percentage of students passing the proficiency test.

TABLE 14-13 An econometrician is interested in evaluating the relation of demand for building materials to mortgage rates in Los Angeles and San Francisco. He believes that the appropriate model is -Referring to Table 14-13, the fitted model for predicting demand in San Francisco 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 average of the percentage of students attending class (% Attendance), average 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: Note: -Referring to Table 14-15, which of the following is the correct alternative hypothesis to test whether daily average of the percentage of students attending class has any effect on percentage of students passing the proficiency test?

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 average of the percentage of students attending class (% Attendance), average 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: Note: -Referring to Table 14-15, there is sufficient evidence that the percentage of students passing the proficiency test depends on all of the 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 average of the percentage of students attending class (% Attendance), average 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: Note: -Referring to Table 14-15, we can conclude that average teacher salary has no impact on the mean percentage of students passing the proficiency test at a 10% level of significance based solely on the 95% confidence interval estimate for β₂.

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: Data for 12 clients on a weight-loss program at the clinic were collected and used to fit the interaction model: Partial output from Microsoft Excel follows: Regression Statistics ANOVA -Referring to Table 14-11, in terms of the β's in the model, give the average change in weight-loss (Y) for every 1 month increase in time in the program (X₁) when attending the morning session.

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: SUMMARY OUTPUT Regression Statistics ANOVA -Referring to Table 14-8, the value of the F-statistic for testing the significance of the entire regression 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 average of the percentage of students attending class (% Attendance), average 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: Note: -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 14-14 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 -Referring to Table 14-14, the fitted model for predicting mileages for 6-cylinder cars 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 average of the percentage of students attending class (% Attendance), average 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: Note: -Referring to Table 14-15, the null hypothesis should be rejected at a 5% level of significance when testing whether daily average of the percentage of students attending class has any effect on percentage of students passing the proficiency test.

In a multiple regression model, which of the following is correct regarding the value of the adjusted r² ?

TABLE 14-6 One of the most common questions of prospective house buyers pertains to the average cost of heating in dollars (Y). To provide its customers with information on that matter, a large real estate firm used the following 4 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit (X₁), the amount of insulation in inches (X₂), the number of windows in the house (X₃), and the age of the furnace in years (X₄). Given below are the EXCEL outputs of two regression models. Model 1 Note: 2.96869E-05 = 2.96869×10^-5 Model 2 Note: 2.9036E-06 = 2.9036×10^-6 -Referring to Table 14-6, what is your decision and conclusion for the test H₀ : β₂ = 0 vs. H₀ : β₂ < 0 at the α = 0.01 level of significance using Model 1?

When an explanatory variable is dropped from a multiple regression model, the adjusted r² can increase.

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 average of the percentage of students attending class (% Attendance), average 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: Note: -Referring to Table 14-15, Referring to Table 14-15, we can conclude that instructional spending per pupil has no impact on the mean percentage of students passing the proficiency test at a 10% level of significance based solely on the 95% confidence interval estimate for β₃.

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. SUMMARY OUTPUT Regression Statistics ANOVA -Referring to Table 14-3, to test whether gross domestic product has a positive impact on consumption, the p-value is