Deck 13: Multiple Regression Analysis

Question 1

In a regression model involving 40 observations, the following estimated regression model was obtained: . For this model, the following statistics are given: SSR = 501 and SSE = 99. What is the value of MSR?

A) 12.525
B) 15
C) 33
D) 167

Answer 1

D

Question 2

Which of the following correctly describes a p-value?

A) It gives the partial change in Y for a unit change in an independent variable, while holding other independent variables constant.
B) It equals the probability ofbeing true, given the claimThe true regression coefficient equals 0.
C) It is an index of the degree of linear association among more than two variables, equal to the square root of the sample coefficient of multiple determination.
D) It equals the ratio of an estimated partial-regression coefficient to its standard error.

Answer 2

B

Question 3

In a multiple regression analysis involving 4 predictor variables and 25 observations, the total sum of squares is 800, and the error sum of squares is 200. What, then, is the value of the F-test statistic for testing the usefulness of this model?

A) 200
B) 50
C) 32
D) 15

Answer 3

D

Question 4

To test the validity of a multiple regression model involving three predictor variables, which of the following is the best formulation of the null hypothesis to be tested?

A)
B)
C)
D)

Question 5

For a multiple regression model, the following statistics are given: Total SS = 500, SSE = 60, and n = 20. In this case, what is the coefficient of determination, expressed as a percentage?

A) 93.81%
B) 88%
C) 77.44%
D) 12%

Question 6

Which of the following correctly describes the coefficient of multiple determination?

A) It is denoted byand is interpreted as the proportion of the total variation that is explained by the regression of y on the predictor variables.
B) It is computed as the ratio of total variation to explained variation.
C) In the case of two independent variables, it describes how well the regression hyperplane fits the data.

Question 7

In multiple regression analysis, what does the ratio MSR/MSE equal?

A) the F-test statistic for testing the usefulness of the regression model
B) the coefficient of determination
C) the adjusted coefficient of determination(adj)
D) the t-test statistic for testing each partial regression coefficient

Question 8

Suppose a regression analysis based on the model with 15 observations produced SSE = 3.55, = 13.131, and = 125.1. In this case, what is the proportion of the total variability in y that is accounted for by and ?

A) 0.0312
B) 0.2704
C) 0.7296
D) 0.9688

Question 9

For a multiple regression model the following statistics are given: Total SS = 400, SSR = 350, k = 4, and n = 20. Given this information, what is the coefficient of determination adjusted for degrees of freedom?

A) 87.5%
B) 84.17%
C) 15.83%
D) 12.5%

Question 10

A three-variable multiple regression establishes an estimated multiple regression equation. Which of the following is a property of that equation?

A) All the estimates derived from it fall on a surface that is called the regression plane.
B) All the estimates derived from it fall on a surface that is called the regression plane, which is positioned among the sample points in such a way as to minimize the sum of the square horizontal deviations between these sample points and their associated estimates.
C) All the estimates derived from it fall on a surface that is called the regression plane, which is positioned among the sample points in such a way as to minimize the sum of the squared horizontal deviations between these sample points and their associated estimates, all of which lie on this plane.

Question 11

In order to test the validity of a multiple regression model involving 4 predictor variables and 25 observations, what are the numerator and denominator degrees of freedom ( , respectively) for the critical value of F?

A) 4 and 20
B) 4 and 25
C) 20 and 4
D) 25 and 4

Question 12

A three-variable multiple regression plane is positioned so as to minimize the sum of squared errors. This sum is given by which of the following expressions?

A)
B)
C)

Question 13

A multiple regression model has the form . As increases by 1 unit, with and held constant, what is expected to happen to y on average?

A) It will increase by 1 unit.
B) It will decrease by 3 units.
C) It will decrease by 4 units.
D) It will increase by 10 units.

Question 14

The test statistic F found in the ANOVA table for testing the usefulness of the regression model is given by which of the following expressions?

A) MSR/MSE
B) MSE/MSR
C) Total SS/SSE
D) SSR/SSE

Question 15

Which of the following is the correct interpretation of the multiple-regression equation ?

A) This equation gives us the estimated valueof the dependent variable for any specified pair of values of the independent variables.
B) The estimated regression coefficientrepresents the slope of the line.
C) The estimated regression coefficientequals the change in estimated Y that is associated with a unit change inand.

Question 16

A multiple regression model involves 5 predictor variables and 30 observations. If we want to test the hypotheses at the 5% significance level, what will the critical value of the rejection region be?

A) 1.699
B) 1.711
C) 2.045
D) 2.064

Question 17

The adjusted multiple coefficient of determination is adjusted for which of the following quantities?

A) the number of regression parameters including
B) the number of dependent variables and the sample size
C) the number of predictor variables and the sample size
D) the correlation coefficient

Question 18

Which of the following correctly describes a multiple-regression ANOVA test?

A) It is a test of the significance of individual coefficients, rather than of the overall significance of the regression model.
B)
B) It amounts to testing the hypothesis that none of the true regression coefficients is 0 and that, therefore, all the independent predictors help explain the variation of the dependent variable.
C) It is a test of the overall significance of a regression, rather than a test of the significance of individual coefficients.
D) Both a and

Question 19

A multiple regression model involves 3 predictor variables and a sample of 20 data points. If we want to test the usefulness of the model at the 1% significance level, what is the critical value of the rejection region?

A) 5.29
B) 5.93
C) 6.30
D) 7.09

Question 20

In a multiple regression analysis involving 24 data points, the mean squares for error, MSE, is 2, and the sum of squares for error, SSE, is 36. Under these circumstances, what must the number of the predictor variables be?

A) 6
B) 5
C) 4
D) 3

Question 21

In a multiple regression model, what is assumed to be true of the standard deviation of the error variable ?

A) It is assumed to be constant for all values of the independent variables.
B) It is assumed to be constant for all values of the dependent variable.
C) It is assumed to be equal to 1.0.
D) It is assumed to be less than 1.0.

Question 22

In multiple regression models, which of the following may be assumed in with respect to the values of the error variable ?

A) They are autocorrelated.
B) They are dependent on each other.
C) They are independent of each other.
D) They are always positive.

Question 23

A multiple regression model has the form . Which of the following is the best interpretation of the coefficient ?

A) It is the change in y per unit change in.
B) It is the change in y per unit change in, holdingconstant.
C) It is the change in y per unit change in, whenandvalues are correlated.
D) It is the change in the average value of y per unit change in, holdingconstant.

Question 24

In testing the validity of a multiple regression model involving 10 independent variables and 100 observations, what will the respective values of the numerator and denominator degrees of freedom for the critical value of F be?

A) 9 and 10
B) 9 and 90
C) 10 and 89
D) 10 and 100

Question 25

In testing the validity of a multiple regression model, a large value of the F-test statistic is indicative of which of the following situations?

A) Most of the variation in the independent variables is explained by the variation in y.
B) Most of the variation in y is explained by the regression equation.
C) Most of the variation in y is unexplained by the regression equation.
D) The model provides a poor fit.

Question 26

To test the validity of a multiple regression model, which of the following tests do we use for the null hypothesis that the regression coefficients are all 0?

A) t test
B) z test
C) F test
D) Rtest

Question 27

Given MSR = 345 and MSE = 431.25, what is the value of the F statistic?

A) 1.25
B) 0.994
C) 0.894
D) 0.800

Question 28

A multiple regression analysis involving 3 independent variables and 25 data points results in a value of 0.769 for the unadjusted multiple coefficient of determination. What is the adjusted multiple coefficient of determination?

A) 0.385
B) 0.591
C) 0.736
D) 0.877

Question 29

A multiple regression model involves five independent variables and a sample of ten data points. If we want to test the validity of the model at the 5% significance level, what is the critical value?

A) 9.36
B) 6.26
C) 4.24
D) 3.33

Question 30

In a simple linear regression , the following pairs of ( ) are given: (6.75, 7.42), (8.96, 8.06), (10.30, 11.65), and (13.24, 12.15). Which of these values is equal to the sum of squares for error?

A) -0.0300
B) 4.2695
C) 39.2500
D) 39.2800

Question 31

What is the range in values of the coefficient of multiple determination?

A) from 1.0 to
B) from 0.0 to 1.0
C) from 1.0 to k, where k is the number of independent variables in the model
D) from 1.0 to n, where n is the number of observations in the dependent variable

Question 32

In a multiple regression model, if the residuals have a constant variance, which of the following should be evident?

A) The residual mean should be close to 0, and the residual standard deviation should be close to 1.0.
B) The plot of residuals against each independent variable should show that the spread in the residuals is about the same at all levels of the independent variables.
C) The residuals should have a variance equal to 0 for all levels of the independent variable.

Question 33

To test the validity of a multiple regression model involving two independent variables, which of the following is the most appropriate null hypothesis?

A)
B)
C)
D)

Question 34

In multiple regression analysis, which of the following is equal to the ratio MSR/MSE?

A) the t-test statistic for testing each individual regression coefficient
B) the F-test statistic for testing the validity of the regression equation
C) the multiple coefficient of determination
D) the adjusted multiple coefficient of determination

Question 35

In order to test the validity of a multiple regression model involving 5 independent variables and 30 observations, what are the respective numerator and denominator degrees of freedom for the critical value of F?

A) 5 and 24
B) 5 and 30
C) 6 and 25
D) 6 and 29

Question 36

Given a multiple regression with a regression sum of squares of 850 and a total sum of square of 1000, what is the coefficient of multiple determination?

A) 0.150
B) 0.387
C) 0.850
D) 0.925

Question 37

A multiple regression equation includes five independent variables, and the coefficient of determination is 0.81. What is the percentage of the variation in y that is explained by the regression equation?

A) 90%
B) 86%
C) 81%
D) about 16%

Question 38

Which of the following best describes the model ?

A) It is a third-order polynomial model.
B) It is a second-order polynomial model.
C) It is a first-order polynomial model since there is only one independent variable x.
D) It is a quadratic model since the termwill often be dropped out of the model based on sample information.

Question 39

For which of the following quantities is the adjusted multiple coefficient of determination adjusted??

A) the number of regression parameters including the y-intercept
B) the number of dependent variables and the sample size
C) the number of independent variables and the sample size
D) the coefficient of correlation and the significance level

Question 40

In multiple regression analysis involving 10 independent variables and 100 observations, how many degrees of freedom will the critical value of t for testing individual coefficients in the model have?

A) 100 degrees of freedom
B) 89 degrees of freedom
C) 10 degrees of freedom
D) 9 degrees of freedom

Question 41

In a regression setting, you should select a model where all the regression coefficients have p-values greater than 0.05.

Question 42

In regression analysis, to what does the term "multicollinearity" refer?

A) the response variables being highly correlated with one another
B) the predictor variables being highly correlated with one another
C) the response variable and the predictor variables being highly correlated with one another
D) the response variables being highly correlated over time

Question 43

In multiple regression analysis, which of the following is a clue that multicollinearity is present?

A) The value ofis large, indicating a good fit, but the individual t tests are nonsignificant.
B) The signs of the regression coefficients are contrary to what we would intuitively expect the contributions of those variables to be.
C) A matrix of correlations, generated by computer, shows which predictor variables are highly correlated with each other and with the response variable y.
D) All of (a), (b), and (c).

Question 44

Which, if any, of the following is an advantage of using stepwise regression compared to just entering all the independent variables at one time?

A) There are no advantages of using stepwise regression over entering all the variables at one time.
B) Stepwise regression allows us to observe the effects of multicollinearity earlier than when all variables are entered at one time.
C) Stepwise regression will generally produce a model with a larger value for the coefficient of determination.

Question 45

Stepwise regression is an iterative procedure that does which of the following?

A) It adds one independent variable at a time.
B) It deletes one independent variable at a time.
C) It deletes one dependent variable at a time.
D) It adds and deletes one independent variable at a time

Question 46

In a regression setting, you should always select the model with the largest .

Question 47

Which of the following methods is used to help assess whether the regression model meets the assumption of having normally distributed residuals?

A) Develop a normal probability plot of the residuals.
B) Develop a histogram of the residuals.
C) both (a) and (b)
D) neither (a) nor (b)

Question 48

In a regression setting, if you add a predictor variable to an existing model, the value of will either remain the same or increase. However, in selecting a model, you should consider the value of the adjusted as well as .

Question 49

How many dummy variables will you need to include if you wish to develop a regression model in which the high school class standing is a qualitative variable with four possible levels of response?

A) 8
B) 5
C) 4
D) 3

Question 50

In stepwise regression procedure, what occurs if two independent variables are highly correlated?

A) Both variables will enter the equation.
B) Only one variable will enter the equation.
C) Neither variable will enter the equation.
D) A third variable equal to their sum will replace them in the equation.

Question 51

Assume you are considering including two additional qualitative variables into a regression model. The first variable has four categories, and the second variable has four categories as well. Given this information, how many indicator variables will be incorporated into the model?

A) 8
B) 7
C) 6
D) 5

Question 52

In multiple regression analysis, which procedure permits variables to enter and leave the model at different stages of its development?

A) forward selection
B) residual analysis
C) backward elimination
D) stepwise regression

Question 53

In a multiple linear regression model, , are independent predictor variables that are measured without error.

Question 54

What may be deduced if multicollinearity exists among the independent variables included in a multiple regression model?

A) The regression coefficients will be difficult to interpret.
B)
B) The standard errors of the regression coefficients for the correlated independent variables will increase
C) The multiple coefficient of determination will assume a value close to 0.
D) Both a and

Question 55

Under which of the following conditions does the of multicollinearity arise?

A) when the dependent variables are highly correlated with one another
B) when the independent variables are highly correlated with one another
C) when the independent variables are highly correlated with the dependent variable

Question 56

In a multiple linear regression model, , is the intercept, while are the partial slopes, or partial regression coefficients.

Question 57

In which of the following situations is stepwise regression especially useful?

A) when there are many independent variables
B) when there are few independent variables
C) when there are many dependent variables
D) when there are few dependent variables

Question 58

In a multiple linear regression model, , the coefficient measures the change in the dependent variable y for a unit change in when all other independent variables are held constant.

Question 59

When the independent variables are correlated with one another in a multiple regression analysis, what is this condition called?

A) heteroscedasticity
B) homoscedasticity
C) multicollinearity
D) causality

Question 60

Typical symptoms of the presence of multicollinearity include which of the following?

A) The estimated regression coefficients vary substantially from sample to sample; this fact raises their standard errors; hence, theis unlikely to be greater than 2, or statistically significant.
B) The estimated regression coefficients change greatly in value as independent variables are dropped from or added to the regression equation.
C) The signs of the estimated regression coefficients are nonsensical; they are negative when common sense suggests positive signs and vice versa.
D) All of the above and more.

Question 61

Multicollinearity is also called collinearity and intercorrelation.

Question 62

A coefficient of multiple correlation, denoted by R, equals the proportion of the total variation in the values of the dependent variable, Y, that is explained by the estimated multiple regression of Y on and possibly additional independent variables ( and so on).

Question 63

Multiple correlation analysis measures the overall strength of association among more than two variables.

Question 64

An estimated partial-regression coefficient gives the partial change in Y for a unit change in that independent variable, while holding other independent variables constant.

Question 65

Multicollinearity exists in virtually all multiple regression models.

Question 66

The y-intercept will usually be negative in a multiple regression model when the regression slope coefficients are positive.

Question 67

Assume that a company is tracking its advertising expenditures as they relate to television ( ) and radio advertising ( ). The owner of the company believes that it would improve the regression model to add a third variable that represents the sum of the advertising on radio and television ( = + ). This assessment is generally correct.

Question 68

The more predictors that are added to a regression model, the larger the coefficient of determination R² value will be.

Question 69

In a multiple regression model, the coefficient of determination will be equal to the square of the largest correlation value between the dependent variable and the independent variables.

Question 70

Having a large number of predictors in a regression model guarantees that the model fit is good.

Question 71

The coefficient of determination, R², represents the proportion of the total variability in y that can be explained by the regression of y on x. When transformed to a percentage, it represents the percentage reduction in the sum of the squares of the error that can be accomplished by using the model to predict the dependent variable as opposed to just using the sample mean of the dependent variable.

Question 72

Multicollinearity does not affect the F-test of the analysis of variance.

Question 73

Multicollinearity is a condition that exists when the independent variables are highly correlated with the dependent variable.

Question 74

The coefficient of multiple determination takes on values between 0 and 1, inclusive.

Question 75

The value will tend to be smaller than the adjusted value when insignificant independent variables are included in the model.

Question 76

An adjusted coefficient of multiple determination, denoted by (adj), is adjusted for the degrees of freedom.

Question 77

In a multiple regression analysis involving six independent variables, the sum of squares are calculated as: Total SS = 900, SSR = 600, and SSE = 300. In this case, the value of the F-test statistic is 150.

Question 78

In a multiple regression model, adding more independent variables that have a low correlation with the dependent variable will decrease the value of the coefficient of multiple determination.

Question 79

If the confidence interval estimate for the regression slope coefficient, based on the sample information, crosses over 0, then the true population regression slope coefficient could be 0.

Question 80

In a multiple regression model, the mean of the residuals is equal to the variance of all combinations of levels of the independent variables.