Deck 15: Multiple Regression and Model Building

Question 1

When the F test is used to test the overall significance of a multiple regression model, if the null hypothesis is rejected, it can be concluded that all of the independent variables are significantly related to the dependent variable y.

Answer 1

False

Question 2

In a multiple regression mode, if the largest variance inflation factor (VIF) is 21.6, then it can be concluded that there are indications of multicollinearity.

Answer 2

True

Question 3

Regression models that employ more than one independent variable are referred to as multiple regression models.

Answer 3

True

Question 4

The assumption of independent error terms in regression analysis is often violated when using time-series data.

Question 5

An application of the multiple regression model generated the following results involving the F test of the overall regression model: p-value = .0012, R² = .67, and s = .076. Thus, the null hypothesis, which states that none of the independent variables are significantly related to the dependent variable, should be rejected at the .05 level of significance.

Question 6

In a multiple regression analysis, if the normal probability plot exhibits approximately a straight line, then it can be concluded that the assumption of normality is not violated.

Question 7

A t-test is used in testing the significance of an individual independent variable.

Question 8

For the same point estimate of the dependent variable and the same level of significance, the confidence interval is always wider than the corresponding prediction interval.

Question 9

Even when an unimportant variable is added to a regression model, the explained variation will increase.

Question 10

Testing the contribution of individual independent variables with t-tests is performed prior to the F test for the model in multiple regression analysis.

Question 11

When the quadratic regression model y = β₀ + β₁x + β₂x² + ε is used, the term β₁ shows the rate of curvature of the parabola.

Question 12

If we are predicting y when the values of the independent variables are x₀₁, x₀₂, . . . , x_0k, the farther the values of x₀₁, x₀₂, . . . , x_0k are from the center of the observed data, the smaller the distance value and the more precise the associated confidence and prediction intervals.

Question 13

In comparing regression models, the regression model with the largest R² will also have the smallest standard error (s).

Question 14

The normal plot is a residual plot that checks the normality assumption.

Question 15

The variance inflation factor (VIF) measures the relationship between the dependent variable and the rest of the independent variables in the regression model.

Question 16

The multiple correlation coefficient can assume any value between zero and 1, inclusive.

Question 17

The error term in the regression model describes the effects of all factors other than the independent variables on y, or the response variable.

Question 18

Because multiple regression models consist of multiple independent variables, residual analysis cannot be performed.

Question 19

In a regression model, a value of the error term depends upon other values of the error term.

Question 20

In a regression model, at any given combination of values of the independent variables, the population of potential error terms is assumed to have an F distribution.

Question 21

It is appropriate to use an interaction variable if the relationship between the dependent variable and one of the independent variables depends on the value of the other independent variable.

Question 22

Backward elimination regression is an automatic model-building procedure.

Question 23

If it is desired to include marital status in a multiple regression model by using the categories single, married, separated, divorced, and widowed, what will be the effect on the model?

A) One more independent variable will be included.
B) Two more independent variables will be included.
C) Three more independent variables will be included.
D) Four more independent variables will be included.
E) Five more independent variables will be included.

Question 24

DFBETAs and DFFITS are statistics used to determine an outlier.

Question 25

In contrast to stepwise regression, backward elimination is an iterative model selection procedure that begins with all potential independent variables and then attempts to remove them one at a time based on the p-value of the independent variable.

Question 26

Leverage value is a statistic used to determine an outlier.

Question 27

A partial F test is used to assess when at least one variable in a subset of squared and interaction variables in the multiple regression model is significant.

Question 28

If a particular multiple regression model has a small value of the C statistic and C for this model is less than k+1, where k is the number of independent variables in the model, then the model should be considered biased and therefore undesirable.

Question 29

The range of feasible values for the multiple coefficient of correlation is from ________.

A) 0 to ∞
B) −1 to 0
C) −1 to 1
D) 0 to 1
E) -∞ to 0

Question 30

Which of the following is not an assumption of the multiple linear regression model?

A) The error terms are independent.
B) The population of error terms has a normal distribution.
C) Populations of error terms observed at different combinations of values of the independent variable (x₁, x₂,. . ., x_k) have equal variances.
D) The level of measurement of the data for the dependent variable is at least ordinal.
E) At any combination of values of x₁, x₂, . . . , x_k, the population of potential error term values has a mean equal to zero.

Question 31

The critical value for DFBETAs statistics is 1.

Question 32

A studentized residual for an observation that is greater than 2 in absolute value is evidence that the observation is an outlier.

Question 33

The range of feasible values for the multiple coefficient of determination is from ________.

A) 0 to ∞
B) −1 to 0
C) −1 to 1
D) 0 to 1
E) -∞ to 0

Question 34

A multiple regression analysis with 20 observations on each of three independent variables and the dependent variable would yield ________ and ________ degrees of freedom, respectively, for regression (explained) and error.

A) 3, 17
B) 3, 16
C) 4, 16
D) 3, 19
E) 3, 20

Question 35

Cook's distance measure is used to detect if an outlier might influence the estimate of a model's parameter.

Question 36

The multiple coefficient of determination is the ________ divided by the total variation.

A) unexplained variation
B) SSE
C) explained variation
D) distance value
E) leverage value

Question 37

Using squared and interaction variables in a multiple regression model results in extreme multicollinearity.

Question 38

The mean square error of a multiple regression model with k independent variables and n observations is ________.

A) SSE / n
B) SSE / [n + (k + 1)]
C) SSE / [n − (k + 1)]
D) SSE / (k + 1)

Question 39

For a given multiple regression model with three independent variables, the value of the adjusted multiple coefficient of determination is________ less than R².

A) always
B) sometimes
C) never

Question 40

When using a multiple regression model, we assume that error terms, or residuals, are distributed according to a(n) ________ distribution.

A) binomial
B) normal
C) exponential
D) Poisson

Question 41

The graph of the prediction equation obtained from the model y = β₀ + β₁X₁ + β₂X₂ + ε is a(n) ________.

A) line
B) plane
C) parabola
D) exponential curve

Question 42

In a multiple regression analysis, if the normal probability plot ________, then it can be concluded that the assumption of normality is not violated.

A) is a straight line
B) has the shape of a symmetric bell-shaped curve
C) is greatly curved
D) is left skewed
E) has the shape of a parabola that opens upward

Question 43

In multiple regression analysis, the mean square regression divided by mean square error yields the ________.

A) standard error
B) F statistic
C) R²
D) adjusted R² or R⁻²
E) t statistic

Question 44

In regression analysis, the standard error(s) is ________ greater than the standard deviation of y, the dependent variable.

A) always
B) sometimes
C) never

Question 45

If we are testing the significance of the independent variable X₁ and we reject the null hypothesis H₀: β₁ = 0, we conclude that

A) X₁ is significantly related to y.
B) X₁ is not significantly related to y.
C) X₁ is an unimportant independent variable.
D) β₁ is significantly related to the dependent variable y.

Question 46

In using the multiple regression method, we can model the effects of the different levels of a qualitative independent variable by using a(n) ________.

A) interaction variable
B) cross-product term
C) quadratic term
D) dummy (indicator) variable
E) variance equalizing transformation

Question 47

An investigator hired by a client suing for sex discrimination has developed a multiple regression model for employee salaries for the company in question. In this multiple regression model, the salaries are in thousands of dollars. For example, a data entry of 35 for the dependent variable indicates a salary of $35,000. The indicator (dummy) variable for gender is coded as X₁ = 0 if male and X₁ = 1 if female. The computer output of this multiple regression model shows that the coefficient for this variable (X₁) is −4.2. The t test showed that X₁ was significant at α = .1. This result implies that for male and female workers of the company,

A) on the average, females earn $4,200 less than males.
B) on the average, males earn $4,200 less than females.
C) on the average, salaries do not differ between males and females.
D) on the average, males have 4.2 more years of experience than females.
E) on the average, females have 4.2 more years of experience than males.

Question 48

Which one of the following is not an assumption about the residuals in a regression model?

A) constant variance
B) independence
C) normality
D) variance of zero
E) mean of zero

Question 49

The multiple ________ measures the proportion of the variation in y, the response variable, explained by the multiple regression model or the set of independent variables included in the multiple regression equation.

A) correlation coefficient
B) coefficient of determination
C) total variation
D) standard error
E) F test

Question 50

Dummy or indicator variables typically are values of zero or one and are used to model the effects of different levels of ________ variables.

A) qualitative
B) quantitative
C) ratio
D) measured

Question 51

The y-intercept, β_0, in a multiple regression model represents the estimated value of the ________ variable, when the value of all independent variables are ________.

A) response, one
B) dummy, zero
C) response, zero
D) dummy, one

Question 52

The ________ term describes the effects on y of all factors other than the independent variables in a multiple regression model.

A) dependent
B) t test
C) error
D) dummy

Question 53

In multiple regression analysis, the explained sum of squares divided by the total sum of squares yields the ________.

A) standard error
B) F statistic
C) R²
D) adjusted R² or R⁻²
E) t statistic

Question 54

The effects of different levels of qualitative independent variables are described using ________ variables.

A) dependent
B) response
C) dummy
D) quantitative

Question 55

Which of the following residual plots is not used in regression analysis?

A) residuals vs. parameter estimates
B) residuals vs. values of an independent variable
C) residuals vs. time order
D) residuals vs. predicted values of the dependent variable
E) standardized residuals vs. predicted values of the dependent variable

Question 56

R² is defined as

A) total variation/explained variation.
B) explained variation/total variation.
C) unexplained variation/explained variation.
D) unexplained variation/total variation.

Question 57

In multiple regression analysis, a desirable residual plot has what type of appearance?

A) curved
B) cyclical
C) fanning out
D) funneling in
E) horizontal band

Question 58

An acceptable residual plot exhibits

A) increasing error variance.
B) decreasing error variance.
C) constant error variance.
D) a curved pattern.
E) a mixture of increasing and decreasing error variance.

Question 59

Which one of the following tools is not used to check the normality of residuals assumption for a multiple regression model?

A) histogram
B) stem-and-leaf display
C) scatter diagram
D) normal plot

Question 60

In multiple regression analysis, which one of the following is the appropriate notation for error, or residual?

A) −

B) −

C) −

D) −

E) None of these answers is correct.

Question 61

The general form of the quadratic multiple regression models is

A) y = β₁x₁ + β₂x₂ + ε
B) y = β₀ + β₁x₁ + β₂x₂ + ε
C) y = β₀ + β₁x + β₂x² + ε
D) y = β₀ + β₁x² + ε
E) y = β₀ + β₁x₁ +β₂x₂²+ ε

Question 62

In using a regression model, if a new independent variable is added, the value of R² (the coefficient of multiple determination) will ________ decrease.

A) always
B) sometimes
C) never

Question 63

As we increase the number of independent variables in a multiple regression model, the F-statistic will ________ increase.

A) always
B) sometimes
C) never

Question 64

In the quadratic regression model y = β₀ + β₁X₁ + β₂X₁² + ε, if the term β₂ is ________ zero, then the parabola opens ________.

A) less than, upward
B) greater than, upward
C) greater than, either upward or downward
D) less than, either upward or downward
E) equal to, downward

Question 65

In a multiple regression analysis, the current model has three independent variables. The analyst decides to add another (fourth) independent variable while retaining the other three independent variables. As a result of this addition, the value of MSE will ________ decrease.

A) always
B) sometimes
C) never

Question 66

In the quadratic regression model y = β₀ + β₁X₁ + β₂X₁² + ε, the β₂ term represents the

A) rate of curvature of the parabola.
B) value of Y when X is zero.
C) shift parameter of the parabola.
D) y-intercept of the parabola.

Question 67

Assumptions of a regression model can be evaluated by plotting and analyzing the ________.

A) independent variables
B) dependent variables
C) error terms
D) beta values

Question 68

The graph of the prediction equation obtained from the model y = β₀ + β₁X₁ + β₂X₁² + ε is a(n) ________.

A) line
B) plane
C) parabola
D) exponential curve

Question 69

If a regression model with k independent variables has a C statistic less than ________, then the model is considered to be desirable.

A) k + 1
B) k
C) k − 1
D) 1/k
E) k − 2

Question 70

The primary use of stepwise regression is to identify the most important ________ that should be included in the multiple regression model.

A) independent variables
B) dependent variables
C) dummy variables
D) quadratic variables

Question 71

In a multiple regression model, the residuals were plotted against the values of one of the independent variables. The plot exhibited a funneling out pattern of residuals. This means that as the value of the independent variable increases, the error terms tend to ________ and the model assumption of ________ is violated.

A) increase, constant variance
B) increase, independence
C) decrease, constant variance
D) decrease, normality

Question 72

Dummy variables take on the values of ________ and are used to model the effects of different levels of qualitative variables.

A) 1 or −1
B) 1 or 2
C) 0 or 2
D) 0 or 1

Question 73

Given the regression model y = β₀ + β₁x₁ + β₂x₂ + β₃x₁² + β₄x₂² + ε, if we wish to test the significance of higher-order terms (x₁² and x₂²), which test would we use?

A) overall F test
B) Durbin-Watson test
C) partial F test
D) t test
E) Cook's distance measure

Question 74

A very insignificant independent variable (an independent variable that has a very weak relationship with the dependent variable) is added to a multiple regression equation. As a result of this change, the value of the explained variation (SSR) will ________, the value of the multiple coefficient of determination (R²) will ________, and the calculated value of the F statistic will most likely ________.

A) decrease, increase, decrease
B) increase, decrease, decrease
C) increase, increase, increase
D) increase, increase, decrease
E) decrease, decrease, decrease

Question 75

If the simple correlation coefficient between two independent variables is greater than .90, then ________ is considered to be severe.

A) autocorrelation
B) interaction
C) multicollinearity
D) coefficient of determination

Question 76

In a multiple regression model, we can conclude that multicollinearity exists if the average variance inflation factor (VIF) is substantially greater than ________.

A) 100
B) 5
C) 10
D) 1

Question 77

In the quadratic regression model y = β₀ + β₁X₁ + β₂X₁² + ε, the β₁ term represents the

A) rate of curvature of the parabola.
B) value of Y when X is zero.
C) shift parameter of the parabola.
D) y-intercept of the parabola.

Question 78

Adding any independent variable to a regression model will increase ________.

A) adjusted R² or R⁻²
B) s
C) MSE
D) R²
E) the length of all prediction intervals

Question 79

Multicollinearity is severe if the largest variance inflation factor (VIF) is greater than ________.

A) 100
B) 5
C) 10
D) 1

Question 80

Plotting the residuals in a time-ordered sequence will reveal possible violations of the ________ of error terms assumption.

A) normality
B) independence
C) constant variation
D) residual sum