Deck 17: Multiple Regression

Question 1

Most statistical software print a second R² statistic, called the coefficient of determination adjusted for degrees of freedom, which has been adjusted to take into account the sample size and the number of independent variables.

Answer 1

True

Question 2

A multiple regression model is assessed to be good if the error sum of squares SSE and the standard error of estimate s _{$\varepsilon$} are both small, the coefficient of determination R² is close to 1, and the value of the test statistic F is large.

Answer 2

True

Question 3

In a multiple regression analysis involving 4 independent variables and 30 data points, the number of degrees of freedom associated with the sum of squares for error, SSE, is 25.

Answer 3

True

Question 4

In regression analysis, the total variation in the dependent variable y, measured by , can be decomposed into two parts: the explained variation, measured by SSR, and the unexplained variation, measured by SSE.

Question 5

In reference to the equation , the value 0.60 is the average change in y per unit change in x₂, regardless of the value of x₁.

Question 6

In a multiple regression analysis involving 50 observations and 5 independent variables, the total variation in y is 475 and SSE = 71.25. Then, the coefficient of determination is 0.85.

Question 7

In multiple regression analysis, when the response surface (the graphical depiction of the regression equation) hits every single point, the sum of squares for error SSE = 0, the standard error of estimate s _{$\varepsilon$} = 0, and the coefficient of determination R² = 1.

Question 8

In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the numerator and denominator degrees of freedom for the critical value of F are 3 and 21, respectively.

Question 9

In reference to the equation , the value 0.12 is the average change in y per unit change in x₁, when x₂ is held constant.

Question 10

A multiple regression equation has a coefficient of determination of 0.81. Then, the percentage of the variation in y that is explained by the regression equation is 90%.

Question 11

In multiple regression, the standard error of estimate is defined by , where n is the sample size and k is the number of independent variables.

Question 12

In reference to the equation , the value -0.80 is the y-intercept.

Question 13

In testing the significance of a multiple regression model with three independent variables, the null hypothesis is .

Question 14

In multiple regression analysis, the adjusted coefficient of determination is adjusted for the number of independent variables and the sample size.

Question 15

One method of diagnosing heteroscedasticity is to plot the residuals against the predicted values of y, then look for a change in the spread of the plotted values.

Question 16

A small value of F indicates that most of the variation in y is explained by the regression equation and that the model is useful.

Question 17

The coefficient of determination R² measures the proportion of variation in y that is explained by the explanatory variables included in the model.

Question 18

When an additional explanatory variable is introduced into a multiple regression model, coefficient of determination adjusted for degrees of freedom can never decrease.

Question 19

The variance of the error variable is required to be constant. When this requirement is violated, the condition is called heteroscedasticity.

Question 20

A multiple regression model involves 40 observations and 4 independent variables produces a total variation in y of 100,000 and SSR = 80,400. Then, the value of MSE is 560.

Question 21

A multiple regression is called "multiple" because it has several explanatory variables.

Question 22

Multicollinearity is a situation in which two or more of the independent variables are highly correlated with each other.

Question 23

A multiple regression model is assessed to be poor if the error sum of squares SSE and the standard error of estimate s _{$\varepsilon$} are both large, the coefficient of determination R² is close to 0, and the value of the test statistic F is large.

Question 24

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

Question 25

One of the consequences of multicollinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.

Question 26

In calculating the standard error of the estimate, , there are (n - k - 1) degrees of freedom, where n is the sample size and k is the number of independent variables in the model.

Question 27

When an explanatory variable is dropped from a multiple regression model, the coefficient of determination can increase.

Question 28

Multicollinearity is present if the dependent variable is linearly related to one of the explanatory variables.

Question 29

The parameter estimates are biased when multicollinearity is present in a multiple regression equation.

Question 30

Multicollinearity is present when there is a high degree of correlation between the independent variables included in the regression model.

Question 31

Because of multicollinearity, the t-tests of the individual coefficients may indicate that some independent variables are not linearly related to the dependent variable, when in fact they are.

Question 32

One of the consequences of multicollinearity in multiple regression is biased estimates on the slope coefficients.

Question 33

When an additional explanatory variable is introduced into a multiple regression model, the coefficient of determination will never decrease.

Question 34

Multicollinearity will result in excessively low standard errors of the parameter estimates reported in the regression output.

Question 35

A high value of the coefficient of determination significantly above 0 in multiple regression, accompanied by insignificant t-statistics on all parameter estimates, very often indicates a high correlation between independent variables in the model.

Question 36

The total variation in y in a regression model will never exceed the regression sum of squares (SSR).

Question 37

Multicollinearity is present when there is a high degree of correlation between the dependent variable and any of the independent variables.

Question 38

From the coefficient of determination, we cannot detect the strength of the relationship between the dependent variable y and any individual independent variable.

Question 39

Multicollinearity affects the t-tests of the individual coefficients as well as the F-test in the analysis of variance for regression because the F-test combines the t-tests into a single test.

Question 40

A multiple regression model has the form . The coefficient b₁ is interpreted as the average change in y per unit change in x₁.

Question 41

The coefficient of determination ranges from:

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

Question 42

A multiple regression model has:

A) only one independent variable.
B) only two independent variables.
C) more than one dependent variable.
D) more than one independent variable.

Question 43

In a multiple regression model, the following statistics are given: SSE = 100, R² = 0.995, k = 5, and n = 15. Then, the coefficient of determination adjusted for degrees of freedom is:

A) 0.930
B) 0.900
C) 0.955
D) 0.855

Question 44

In a multiple regression model, the mean of the probability distribution of the error variable $\varepsilon$ is assumed to be:

A) k, where k is the number of independent variables included in the model.
B) 1.0
C) 0.0
D) None of these choices.

Question 45

In a multiple regression analysis, if the model provides a poor fit, this indicates that:

A) the coefficient of determination will be close to zero.
B) the standard error of estimate will be large.
C) the sum of squares for error will be large.
D) All of these choices are true.

Question 46

The least squares method requires that the variance of the error variable $\varepsilon$ is a constant no matter what the value of x is. When this requirement is violated, the condition is called:

A) heteroscedasticity.
B) homoscedasticity.
C) influential observation.
D) non-independence of $\varepsilon$ .

Question 47

In multiple regression analysis, the ratio MSR/MSE yields the:

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

Question 48

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

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

Question 49

For a multiple regression model the following statistics are given: Total variation in y = 250, SSE = 50, k = 4, and n = 20. Then, the coefficient of determination adjusted for the degrees of freedom is:

A) 0.800
B) 0.747
C) 0.840
D) 0.775

Question 50

A multiple regression model has the form . As x₃ increases by one unit, with x₁ and x₂ held constant, the y on average is expected to:

A) increase by 1 unit.
B) increase by 12 units.
C) decrease by 4 units.
D) decrease by 16 units.

Question 51

A multiple regression model involves 10 independent variables and 30 observations. If we want to test at the 5% significance level whether one of the coefficients is = 0 (vs. $\neq$ 0) the critical value will be:

A) 2.228
B) 2.093
C) 1.729
D) 1.697

Question 52

In a multiple regression analysis involving k independent variables and n data points, the number of degrees of freedom associated with the sum of squares for error is:

A) k - 1
B) n - k
C) n - 1
D) n -k - 1

Question 53

A multiple regression model involves 5 independent variables and a sample of 10 data points. If we want to test the validity of the model at the 5% significance level, the critical value is:

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

Question 54

For a multiple regression model, the following statistics are given: Total variation in y = 500, SSE = 80, and n = 25. Then, the coefficient of determination is:

A) 0.84
B) 0.16
C) 0.3125
D) 0.05

Question 55

A multiple regression model has the form: . As x₂ increases by one unit, holding x₁ constant, then the value of y will increase by:

A) 7.25 units
B) 6 units on average
C) 2 units
D) None of these choices

Question 56

The adjusted coefficient of determination is adjusted for the:

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

Question 57

In a multiple regression analysis involving 6 independent variables, the total variation in y is 900 and SSR = 600. What is the value of SSE?

A) 300
B) 1.50
C) 0.67
D) None of these choices.

Question 58

A multiple regression analysis involving three independent variables and 25 data points results in a value of 0.769 for the unadjusted coefficient of determination. Then, the adjusted coefficient of determination is:

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

Question 59

To test the validity of a multiple regression model, we test the null hypothesis that the regression coefficients are all zero by applying the:

A) F-test
B) t-test
C) z-test
D) None of these choices.

Question 60

Suppose a multiple regression analysis involving 25 data points has and SSE = 36. Then, the number of the independent variables must be:

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

Question 61

In a multiple regression model, the value of the coefficient of determination has to fall between

A) -1 and +1.
B) 0 and +1.
C) -1 and 0.
D) None of these choices.

Question 62

For a multiple regression model, the total variation in y can be expressed as:

A) SSE - SSR.
B) SSR -SSE.
C) SSR + SSE.
D) SSR / SSE.

Question 63

In a multiple regression model, the error variable $\varepsilon$ is assumed to have a mean of:

A) -1.0
B) 0.0
C) 1.0
D) None of these choices.

Question 64

Multiple regression has four requirements for the error variable. One is that the probability distribution of the error variable is ____________________.

Question 65

Which of the following statements regarding multicollinearity is not true?

A) It exists in virtually all multiple regression models.
B) It is also called collinearity and intercorrelation.
C) It is a condition that exists when the independent variables are highly correlated with the dependent variable.
D) All of these choices are true.

Question 66

A multiple regression equation includes 5 independent variables, and the coefficient of determination is 0.81. The percentage of the variation in y that is explained by the regression equation is:

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

Question 67

In a multiple regression analysis, there are 20 data points and 4 independent variables, and the sum of the squared differences between observed and predicted values of y is 180. The standard error of estimate will be:

A) 9.000
B) 6.708
C) 3.464
D) 3.000

Question 68

When the error variable does not have constant variance, this condition is called ____________________.

Question 69

In a multiple regression model, the probability distribution of the error variable $\varepsilon$ is assumed to be:

A) normal.
B) non-normal.
C) positively skewed.
D) negatively skewed.

Question 70

In a multiple regression analysis involving 40 observations and 5 independent variables, the following statistics are given: Total variation in y = 350 and SSE = 50. Then, the coefficient of determination is:

A) 0.8408
B) 0.8571
C) 0.8469
D) 0.8529

Question 71

A multiple regression model has the form . The coefficient b₁ is interpreted as the change in the average value of y per unit change in ________ holding ________ constant.

Question 72

If multicollinearity exists among the independent variables included in a multiple regression model, then:

A) the regression coefficients will be difficult to interpret.
B) the standard errors of the regression coefficients for the correlated independent variables will increase.
C) one or more of the coefficients may have the wrong sign.
D) All of these choices are true.

Question 73

For the following multiple regression model: , a unit increase in x₁, holding x₂ and x₃ constant, results in:

A) a decrease of 3 units on average in the value of y.
B) an increase of 8 units in the value of y.
C) an increase of 3 units on average in the value of y.
D) None of these choices.

Question 74

The problem of multicollinearity arises when the:

A) dependent variables are highly correlated with one another.
B) independent variables are highly correlated with one another.
C) independent variables are highly correlated with the dependent variable.
D) None of these choices.

Question 75

The coefficient of determination ____________________ for degrees of freedom takes into account the sample size and the number of independent variables when assessing model fit.

Question 76

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

A) heteroscedasticity.
B) homoscedasticity.
C) multicollinearity.
D) None of these choices.

Question 77

For the multiple regression model: , if x₂ were to increase by 5, holding x₁ and x₃ constant, the value of y will:

A) increase by 5.
B) increase by 75.
C) decrease on average by 5.
D) decrease on average by 75.

Question 78

The validity of a multiple regression model is tested using a(n) _________ test.

Question 79

We test an individual coefficient in a multiple regression model using a(n) _________ test.

Question 80

If a group of independent variables are not significant individually but are significant as a group at a specified level of significance, this is most likely due to:

A) heteroscedasticity.
B) an error in the analysis.
C) multicollinearity.
D) None of these choices.