Deck 14: Multiple Regression Analysis

Question 1

The interaction between two variables x₁ and x₂ can be modeled by including the predictor variable β₁β₂ into the multiple regression model.

Answer 1

False

Question 2

When fitting a multiple regression model is desirable to have both a large value of R² and a small value of s_e.

Answer 2

True

Question 3

SSRegr + SSResid = SSTo.

Answer 3

True

Question 4

Two variables x₁ and x₂ are said to interact when the change in the mean value of y associated with a one unit increase in one variable depends on the value of the other variable.

Question 5

In the general multiple regression model, y = α + β₁x₁ + β₂x₂ + ... + βkxk + e, each observation in the sample consists of k + 1 numbers.

Question 6

The standard deviation of used in a prediction interval is .

Question 7

The estimated regression function is used for both estimating the mean value of y and predicting the value of y when x₁, x₂, ..., xk are fixed values.

Question 8

The predicted values, the residuals and SSResid for a multiple regression model are interpreted as they were for the simple linear regression model.

Question 9

The alternative hypothesis in the model utility test is Ha : none of the β's are 0.

Question 10

A variable taking on only the values 0 and 1 is called a dummy or indicator variable.

Question 11

The least squares estimate of β₁ is an unbiased statistic for estimating βi.

Question 12

A complete second order regression model in the variables x₁ and x₂ is given by y = α + β₁x₁ + β₂x₂ + β₃x₁² + β₄x₂² +
e.

Question 13

It is possible to carry out the model utility test of H₀ : β₁ = β₂ = ... = βk = 0 when only R², n, and k are known.

Question 14

In a multiple regression model, the utility of the model can be tested with a t test.

Question 15

The polynomial regression model used to fit a parabolic pattern in the observed data is y = α + β₁x + β₂x² +
e.

Question 16

In the quadratic regression model, y = α + β₁x + β₂x² + e, β₂ can be interpreted as the amount that y will be expected to change when the value of x is increased by one unit.

Question 17

A general additive multiple regression model has the form y = α + β₁x₁ + β₂x₂ + ... + βkxk.

Question 18

In the quadratic regression model, y = α + β₁x + β₂x² + e, the parabola opens upward when β₂ > 0 and downward when β₂ < 0.

Question 19

In the general additive regression model, the mean value of y for fixed x₁, x₂, ... xk values is α + β₁ + β₂ + ... + βk.

Question 20

The value of adjusted R² is always smaller than the value of R².

Question 21

A Senate committee is studying the cost of health care and is interested in the relationship between y = the monthly premium paid, x₁ = the age of the policy holder, and x₂ = the number of dependents on the policy. Partial computer output for the Senate study is given below. The regression equation is Prem = 177 + 1.3 x₁ + 68 x₂ ​ S = 17.25 R-sq = 59.1% ​ Use a significance level of .05 for all tests requested.
a) Calculate and interpret a 95% confidence interval for β₂.
b) Does the model appear to be useful? Test the relevant hypothesis.
c) Conduct a test for the following pair of hypotheses: H₀ : β₁ = 0 vs. β₁ ≠ 0.
d) Based on your result in part (c), would you conclude that the age of the policy holder is an important variable? Explain your reasoning.
e) An estimate of the mean monthly premium for a policy holder 23 years old with 2 dependents is desired. Compute a 90% confidence interval for α + β₁(23) + β₂(2) if the estimated standard deviation of a + b₁(23) + b₂(2) is 35. Interpret the resulting interval.
f) A single individual, 23 years old with 2 dependents, is identified. Predict the monthly premium for this person using a 90% interval.

Question 22

Briefly explain what it means when two variables are said to interact.

Question 23

Estimate the P-value for the model utility F test given that k = 4, n = 27, calculated F = 2.81. ​

A) P-value < 0.001
B) 0.001 < P-value < 0.01
C) 0.01 < P-value < 0.05
D) 0.05 < P-value < 0.1
E) P-value > 0.1

Question 24

Exhibit 14-1 To comply with recent Federal legislation, school districts must study their students' growth as a whole, as well as the achievement of various subgroups of students. Over a 3-day period, students are assessed on their reading achievement, science and math knowledge, and social studies skills and these results are combined into a global "composite" score. To analyze the increase in this global score from the Freshman year to the Sophomore year, the model y = α + β1x1 + β2x2 + e was fit to a sample of student data. (The actual data contained categorical variables for each ethnic subgroup. To simplify the analysis, only the African American / White categorical variable is included here.)
y = growth in composite score (Soph. - Fresh. score)
x₁ = last year's composite score
x₂ = 1 if a student is African American, 0 if white
x₃ = 1 if a student receives free or reduced price lunch (a measure of socio-economic status), 0 if not
The computer output from the regression analysis is shown below.



The data in Exhibit 14-1 were reanalyzed after adding an interaction variable, AAFR, where x₄ = x₂x₃. The computer output is shown below: ​ ​ ​
a) Is the Model Utility test significant at the .10 level? Explain your reasoning, referring to specific information in the computer output.
b) Calculate the expected mean growth in composite score for African-American students who scored 280 as Freshmen and are receiving free/reduced lunch.
c) One concern the Federal legislation is intended to address are differences in the school's impact on disadvantaged youngsters. Using the computer output from the interaction model, what, if any, differences in growth scores on the Composite Score from the Freshman to Sophomore year are statistically significant at the .05 level? Does it appear that the different subgroups have different amounts of growth? Justify your reasoning with appropriate references to the data analysis presented in the computer output.

Question 25

Journalists are trying to find out the main factors used by a railway company in establishing dynamic pricing for tickets. One of the possible regression models, based on the information on 20 tickets, is ​ y = Price for a railway ticket ($) ​ x₁ = Number of days before departure ​ x₂ = Weekday (1 = Fri/Sat/Sun, 0 = Mon/Tue/Wen/Thu) ​ x₃ = Distance (km) ​ x₄ = Demand (1 to 5 scale) ​ Using the Minitab output results from fitting this model, carry out the model utility test at a 0.05 significance level. ​ ​

A) P-value < 0.05, there appears to be a useful linear relationship between y and at least one of the three predictors.
B) P-value < 0.05, there appears to be a useful linear relationship between y and each of the three predictors.
C) P-value > 0.05, there is no useful linear relationship between y and any of the predictors.
D) P-value > 0.05, there is a useful linear relationship between y and each of the three predictors.
E) P-value > 0.05, there is no useful linear relationship between y and at least one of the predictors.

Question 26

The cost of renting premises consists of a plurality of parameters. A real estate company attempts to identify the most significant factors and proposes a multiple regression model based on a sample of n = 18 observations. ​ y = Monthly rent ($) ​ x₁ = Surface area (m²) ​ x₂ = Historic building (1 = yes, 0 = no) ​ x₃ = Prestige of a district (1 to 5 scale) ​ x₄ = Parking facilities (1 = yes, 0 = no) ​ x₅ = Availability of infrastructure (1 to 5 scale) Suppose that SSRegr = 846,325 and SSTo = 3,900,000. Calculate the values of R² and adjusted R². Explain the difference between them. ​

A) Adjusted R² is significantly smaller than R², because R² itself is rather small.
B) Adjusted R² is significantly smaller than R², because the number of predictors is large as compared to the number of observations.
C) Adjusted R² is significantly larger than R², because R² itself is rather small.
D) Adjusted R² is significantly greater than R², because the number of predictors is large as compared to the number of observations.
E) The difference between adjusted R² and R² is not significant, because R² is substantial and there are only a few predictors.

Question 27

The owners of an online shop use a multiple regression model with three independent variables where ​ y = Shipping Cost ($) ​ x₁ = Product weight (kg) ​ x₂ = Number of items ​ x₃ = Total amount ($) ​ The regression model is ​ y = 8.50 + 12x₁ + 0.1x₂ + 0.05x₃ + 0.08x₂x₃ + e ​ Interpret the value of β₁ for this model. ​

A) The average change in shipping costs associated with an increase in the number of items by 1 when the product weight and the total amount are held fixed.
B) The average change in shipping costs associated with a 1-dollar increase in the total amount when the product weight and the number of items are held fixed.
C) The difference between the slopes of regression lines depending on the total amount and the number of items.
D) The average change in shipping costs associated with a 1-kg increase in product weight when the number of items and the total amount are held fixed.
E) The average change in shipping costs when the perceived increase value of the number of items by 1 unit depends on the change of the total amount.

Question 28

Consider a regression analysis with four independent variables x₁, x₂, x₃, and x₄. Select the equation for the regression model that includes all independent variables as predictors, two interaction terms, and one quadratic term. ​

A)
B)
C)
D)
E)

Question 29

Exhibit 14-1 To comply with recent Federal legislation, school districts must study their students' growth as a whole, as well as the achievement of various subgroups of students. Over a 3-day period, students are assessed on their reading achievement, science and math knowledge, and social studies skills and these results are combined into a global "composite" score. To analyze the increase in this global score from the Freshman year to the Sophomore year, the model y = α + β1x1 + β2x2 + e was fit to a sample of student data. (The actual data contained categorical variables for each ethnic subgroup. To simplify the analysis, only the African American / White categorical variable is included here.)
y = growth in composite score (Soph. - Fresh. score)
x₁ = last year's composite score
x₂ = 1 if a student is African American, 0 if white
x₃ = 1 if a student receives free or reduced price lunch (a measure of socio-economic status), 0 if not
The computer output from the regression analysis is shown below.



Refer to Exhibit 14-1. a) Is the Model Utility test significant at the .10 level? Explain your reasoning, referring to specific information in the computer output.
b) Calculate the estimated mean growth in composite score for white students who scored 280 as Freshmen and who are receiving free/reduced lunch.

Question 30

Including an additional predictor to a multiple regression model will always increase the value of the adjusted R² value.

Question 31

Multicollinearity is a model selection procedure that can be used to compare different models.

Question 32

The Carolina Reaper pepper is considered one of the hottest peppers in the world. However, manufacturers of sauces made from Carolina Reaper often overstate the pungency given on the package to separate their product and ensure sales. An independent laboratory provided a multiple regression model for determining the pungency of the sauce based on a sample of n = 20 sauces. ​ y = Pungency of a sauce (Scoville heat units) ​ x₁ = Carolina Reaper content (%) ​ x₂ = Sugar content (%) ​ x₃ = Vinegar content (%) ​ The estimated regression function was and R² = 0.979. Does the result of a model utility test at α = 0.01 indicate that this multiple regression model is useful? Assume that the random deviation distribution is normal. ​

A) P-value < 0.001, there appears to be a useful linear relationship between y and each of the three predictors.
B) P-value > 0.01, there is no useful linear relationship between y and any of the predictors.
C) P-value < 0.001, there appears to be a useful linear relationship between y and at least one of the three predictors.
D) 0.001< P-value < 0.01, there is a useful linear relationship between y and each of the three predictors.
E) P-value > 0.01, there is no useful linear relationship between y and at least one of the predictors.

Question 33

Briefly explain how to interpret the value of β₁ in the model y = α + β₁x₁ + β₂x₂ + e, when the variables x₁ and x₂ are independent.

Question 34

How do the R² and adjusted R² differ?

Question 35

The president of a manufacturing firm used a regression model of the form y = α + β₁x₁ + β₂x₂ + β₃x₃ + e to study the relationship between the variables
y = production cost (in dollars)
x₁ = machine time to produce one unit of the product (in minutes)
x₂ = material and labor costs per unit (in dollars)
x₃ = percentage of defective products produced All of seven of the possible models containing combinations of these three variables were fit resulting in the summary table below ​
a) If n = 20, compute the value of the adjusted R² statistic for each of the 7 models.
b) Which of the single variable models is the best? Explain your choice in a few sentences.
c) Let model A be the model with variables x₁, x₂, and model B be the model with variables x₁, x₂, x₃. Which of these two models appears to be the better model? Explain your choice in a few sentences.
d) Which of the seven models appears to be the best overall model? Explain your choice in a few sentences.

Question 36

The largest R² for any two-predictor model is always less than or equal to the largest R² for any three-predictor model.

Question 37

A producer of stainless steel products presented data on y = Workshop expenses (thousands of dollars) as a function of x₁ = Amount of raw material (tons), x₂ = Number of hours worked (hundreds), and x₃ = Quantity of production (thousands of units). Suppose that there is an interaction between the amount of raw material and the quantity of production. What additional predictor variable should be added to the model? ​

A)
B)
C)
D)
E)

Question 38

A normal probability plot of the standardized residuals can be used to investigate whether it is plausible that the distribution of e is approximately normal.