Deck 19: Multiple Regression

Question 1

To test the validity of a multiple regression model involving 2 independent variables, the null hypothesis is that:

A) $\beta$ ₀ = $\beta$ ₁ = $\beta$ ₂.
B) $\beta$ ₀ = $\beta$ ₁ = $\beta$ ₂ = 0.
C) $\beta$ ₁ = $\beta$ ₂ = 0.
D) $\beta$ ₁ = $\beta$ ₂.

Answer 1

$\beta$ ₁ = $\beta$ ₂ = 0.

Question 2

The adjusted multiple coefficient of determination is adjusted for the:

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

Answer 2

C

Question 3

Which of the following best explains a small F-statistic when testing the validity of a multiple regression model?

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

Answer 3

D

Question 4

In a multiple regression analysis involving 6 independent variables and a sample of 19 data points the total variation in y is SS_y = 900 and the amount of variation in y that is explained by the variations in the independent variables is SSR = 600. The value of the F-test statistic for this model is:

A) 4.0.
B) 4.3.
C) 4.8.
D) 6.3.

Question 5

In order to test the validity of a multiple regression model involving 4 independent variables and 35 observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are:

A) 4 and 35.
B) 3 and 32.
C) 3 and 34.
D) 4 and 30.

Question 6

A multiple regression model involves8 independent variables and 32 observations. If we want to test at the 5% significance level the parameter $\beta _ { 4 }$ , the critical value will be:

A) 1.714.
B) 2.042.
C) 2.064.
D) 2.069.

Question 7

Which of the following best describes the ratio MSR/MSE in a multiple linear regression model?

A) Sum of squares of the residuals.
B) t-test statistic to test the individual regression coefficients of the independent variables.
C) F-test to test the overall significance of the regression model.
D) Adjusted multiple coefficient of determination.

Question 8

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. The adjusted multiple coefficient of determination is:

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

Question 9

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

A) 1.0.
B) 0.0.
C) any value greater than 1.
D) k, where k is the number of independent variables included in the model.

Question 10

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 regression is:

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

Question 11

In multiple regression models, the values of the error variable $\varepsilon$ are assumed to be:

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

Question 12

An estimated multiple regression model has the form $\hat{y}$ = 8 + 3x₁ -5x₂ -4x₃. As x₁ increases by 1 unit, with x₂ and x₃ held constant, the value of y, on average, is estimated to:

A) decrease by 3 unit.
B) increase by 3 units.
C) decrease by 6 units.
D) increase by 11 units.

Question 13

A multiple regression model involves 5 independent variables and the sample size is 30. If we want to test the validity of the model at the 5% significance level, the critical value is:

A) 2.59.
B) 2.53.
C) 2.62.
D) 2.56.

Question 14

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

A) constant for all values of the independent variables.
B) constant for all values of the dependent variable.
C) 1.0.
D) None of these choices are correct.

Question 15

Which of the following is used to test the significance of the overall regression equation?

A) t-test..
B) z-test.
C) χ² test
D) F-test

Question 16

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

A) multicollinearity.
B) homoscedasticity.
C) heteroscedasticity.
D) linearity.

Question 17

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) dependent variables are highly correlated with one another or independent variables are highly correlated with one another..

Question 18

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

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

Question 19

Which of the following statements is not true?

A) Multicollinearity exists in virtually all multiple regression models.
B) Multicollinearity is also called collinearity and intercorrelation.
C) Multicollinearity is a condition that exists when the independent variables are highly correlated with the dependent variable.
D) Multicollinearity does not affect the F-test of the analysis of variance.

Question 20

In a multiple regression analysis, when there is no linear relationship between each of the independent variables and the dependent variable, then:

A) multiple t-tests of the individual coefficients will likely show some are significant.
B) we will conclude erroneously that the model has some validity.
C) the chance of erroneously concluding that the model is useful is substantially less with the F-test than with multiple t-tests.
D) All of these choices are correct.

Question 21

In a multiple regression analysis involving 40 observations and 5 independent variables, total variation in y = SSY = 350 and SSE = 50. The multiple coefficient of determination is:

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

Question 22

For the estimated multiple regression model $\hat{y}$ = 30 - 4x₁ + 5x₂ +3 x₃, a one unit increase in x₃, holding x₁ and x₂ constant, will result in which of the following changes in y?

A) y will increase by 3 units.
B) y will increase by 2 units, estimated, on average.
C) y will increase by 33 units
D) y will increase by 3 units, estimated, on average.

Question 23

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) any value smaller than -1.0.

Question 24

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 25

Which of the following best describes a multiple linear regression model?

A) A multiple linear regression model has more than one dependent variable.
B) A multiple linear regression model has more than one independent variable.
C) A multiple linear regression model must have more than one independent variable and more than one independent variable.
D) A multiple linear regression model has one independent variable.

Question 26

Which of the following best describes the range of the coefficient of multiple determination?

A) 0 to 1
B) −1 to 1
C) 1.0 to n, where n is the number of observations in the sample.
D) 1 to k, where k is the number of independent variables in the model.

Question 27

The multiple coefficient of determination is defined as:

A) SSE/SSY.
B) MSE/MSR.
C) 1 - (SSE/SSY).
D) 1 - (MSE/MSR).

Question 28

The graphical depiction of the equation of a multiple regression model with k independent variables (k > 1) is referred to as:

A) a straight line.
B) the response variable.
C) the response surface.
D) a plane only when k = 3.

Question 29

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

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

Question 30

Which of the following is not true when we add an independent variable to a multiple regression model?

A) The adjusted coefficient of determination can assume a negative value.
B) The unadjusted coefficient of determination always increases.
C) The unadjusted coefficient of determination may increase or decrease.
D) The adjusted coefficient of determination may increase.

Question 31

Which of the following measures can be used to assess a multiple regression model's fit?

A) The sum of squares for error.
B) The sum of squares for regression.
C) The standard error of estimate.
D) A single t-test.

Question 32

In a multiple regression model, the following statistics are given: SSE = 100, $R ^ { 2 } = 0.995$ , k = 5, n = 15.
The multiple coefficient of determination adjusted for degrees of freedom is:

A) 0.955.
B) 0.992.
C) 0.930.
D) None of these choices are correct.

Question 33

In testing the validity of a multiple regression model involving 5 independent variables and 30 observations, the numbers of degrees of freedom for the numerator and denominator (respectively) for the critical value of F will be:

A) 5 and 25.
B) 24 and 5.
C) 25 and 5.
D) 5 and 24.

Question 34

For a multiple regression model:

A) SSY = SSR - SSE.
B) SSE = SSR - SSY.
C) SSR = SSE - SSY.
D) SSY = SSE + SSR.

Question 35

For a multiple regression model with n = 35 and k = 4, the following statistics are given: SS_y = 500 and SSE = 100. The coefficient of determination is:

A) 0.82.
B) 0.80.
C) 0.77.
D) 0.20.

Question 36

An estimated multiple regression model has the form ŷ = 100 − 2x₁ + 9x₂. As x₁ increases by 1 unit while holding x₂ constant, which of the following best describes the change in y?

A) y will increase by 2 units, estimated, on average.
B) y will decrease by 98 units, estimated, on average.
C) y will increase by 98 units, estimated, on average.
D) y will decrease by 2 units, estimated, on average.

Question 37

If none of the data points for a multiple regression model with two independent variables were on the regression plane, then the multiple coefficient of determination would be:

A) -1.0.
B) 1.0.
C) any number between -1 and 1, inclusive.
D) any number greater than or equal to zero but smaller than 1.

Question 38

A multiple regression model has the form ŷ = b₀ + b₁x₁ + b₂x₂. Which of the following best describes b₂?

A) We estimate for each one unit increase in x₂, that y will increase by b₂ units, on average.
B) Whilst holding x₁ constant, we estimate for each one unit increase in x₂, that y will increase by b₂ units, on average.
C) Whilst holding x₂ constant, we estimate for each one unit increase in x₁, that y will increase by b₂ units, on average.
D) Whilst holding x₁ constant, we estimate for each one unit increase in x₂, that y will increase by b₁ units, on average.

Question 39

In a multiple regression analysis involving 20 observations and 5 independent variables, total variation in y = SSY = 250 and SSE = 35. The multiple coefficient of determination, adjusted for degrees of freedom, is:

A) 0.810.
B) 0.860.
C) 0.835.
D) 0.831.

Question 40

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

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

Question 41

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of $b _ { 2 }$ , this represents the probability that:

A) $\left| b _ { 2 } \right|$ could be this large if Ho: $\beta _ { 2 } \neq 0$ is really true
B) $\left| b _ { 2 } \right|$ could be this large if Ho: $\beta _ { 2 } = 0$ is really false
C) $\left| b _ { 2 } \right|$ could be this large if Ho: $\beta _ { 2 } \neq 0$ is really true.
D) $\left| b _ { 2 } \right|$ could be this large if Ho: $\beta _ { 2 } = 0$ is really true.

Question 42

For the multiple regression model $= 75 + 25 x _ { 1 } - 15 x _ { 2 } + 10 x _ { 3 }$ , if $x _ { 2 }$ were to increase by 5, holding $x _ { 1 }$ and $x _ { 3 }$ constant, the value of y would:

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

Question 43

Given the multiple linear regression equation $= - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value -0.80 is the $y$ intercept.

Question 44

For each x term in the multiple regression equation, the corresponding $\beta$ is referred to as a partial regression coefficient or slope of the independent variable.

Question 45

For a set of 30 data points, Excel has found the estimated multiple regression equation to be $\hat{y}$ = -8.61 + 22x₁ + 7x₂ + 28x₃, and has listed the t statistic for testing the significance of each regression coefficient. Using the 5% significance level for testing whether $\beta$ ₃ = 0, the critical region will be that the absolute value of the t statistic for $\beta$ ₃ is greater than or equal to:

A) 2.056.
B) 2.045.
C) 1.703.
D) 1.706.

Question 46

A multiple regression equation includes 5 independent variables, and the coefficient of determination is 0.64. The percentage of the variation in y that is explained by the regression equation is:

A) 8%.
B) 12.8%.
C) 41%.
D) 64%

Question 47

A multiple regression analysis that includes 20 data points and 4 independent variables results in total variation in y = SSY = 200 and SSR = 160. The multiple standard error of estimate will be:

A) 0.80.
B) 3.266.
C) 3.651.
D) 1.633.

Question 48

A multiple regression analysis that includes 4 independent variables results in a sum of squares for regression of 1200 and a sum of squares for error of 800. The multiple coefficient of determination will be:

A) 0.667.
B) 0.600.
C) 0.400.
D) 0.200.

Question 49

In a regression model involving 60 observations, the following estimated regression model was obtained: $= 51.4 + 0.70 x _ { 1 } + 0.679 x _ { 2 } - 0.378 x _ { 3 }$ For this model, total variation in y = SSY = 119,724 and SSR = 29,029.72. The value of MSE is:

A) 1619.541.
B) 9676.572.
C) 1995.400.
D) 5020.235.

Question 50

In testing the validity of a multiple regression model in which there are four independent variables, the null hypothesis is:

A) $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 1$ .
B) $H _ { 0 } : \beta _ { 0 } = \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 }$ .
C) $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 0$ .
D) $H _ { 0 } : \beta _ { 0 } = \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } \neq 0$ .

Question 51

In multiple regression, the descriptor 'multiple' refers to more than one independent variable.

Question 52

In a regression model involving 50 observations, the following estimated regression model was obtained: ŷ = 10.5 + 3.2x₁ + 5.8x₂ + 6.5x₃. For this model, SSR = 450 and SSE = 175. The value of MSE is:

A) 9.783.
B) 58.333.
C) 150.000.
D) 3.804.

Question 53

In a regression model involving 30 observations, the following estimated regression model was obtained: $= 60 + 2.8 x _ { 1 } + 1.2 x _ { 2 } - x _ { 3 }$ . For this model, total variation in y = SSY = 800 and SSE = 200. The value of the F-statistic for testing the validity of this model is:

A) 26.00.
B) 7.69.
C) 3.38.
D) 0.039.

Question 54

In a multiple regression analysis involving 25 data points and 5 independent variables, the sum of squares terms are calculated as: total variation in y = SSY = 500, SSR = 300, and SSE = 200. In testing the validity of the regression model, the F-value of the test statistic will be:

A) 5.70.
B) 2.50.
C) 1.50.
D) 0.176.

Question 55

In multiple regression analysis involving 9 independent variables and 110 observations, the critical value of t for testing individual coefficients in the model will have:

A) 109 degrees of freedom.
B) 8 degrees of freedom.
C) 99 degrees of freedom.
D) 100 degrees of freedom.

Question 56

Which of the following best describes the Durbin-Watson test?

A) The Durbin-Watson test is used to determine if there is multicollinearity.
B) The Durbin-Watson test is used to determine if there is heteroscedasticity.
C) The Durbin-Watson test is used to determine if there is first-order autocorrelation.
D) The Durbin-Watson test is used to determine if there is homoscedasticity.

Question 57

Which of the following best describes first-order autocorrelation?

A) First-order autocorrelation is a condition in which there is no relationship between consecutive residuals.
B) First-order autocorrelation is a condition in which the data is skewed.
C) First-order autocorrelation is a condition in which consecutive residuals differ greatly.
D) First-order autocorrelation is a condition in which a relationship exists between consecutive residuals.

Question 58

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 multiple standard error of estimate will be:

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

Question 59

Multiple linear regression is used to estimate the linear relationship between one dependent variable and more than one independent variables.

Question 60

Multicollinearity is a situation in which the independent variables are highly correlated with the dependent variable.

Question 61

In regression analysis, we judge the magnitude of the standard error of estimate relative to the values of the dependent variable, and particularly to the mean of y.

Question 62

A multiple regression model involves 40 observations and 4 independent variables produces
SST = 100 000 and SSR = 82,500. The value of MSE is 500.

Question 63

In reference to the equation $= 1.86 - 0.51 x _ { 1 } + 0.60 \quad x _ { 2 }$ , the value 0.60 is the change in $y$ per unit change in $x _ { 2 }$ , regardless of the value of $x _ { 1 }$ .

Question 64

In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is $= 72 + 3.2 x _ { 1 } + 1.5 x _ { 2 } - x _ { 3 }$ . For this model, SST = 800 and SSE = 245. The value of the F-statistic for testing the significance of this model is 15.102.

Question 65

A multiple regression model has the form $\hat{y}$ = 24 - 0.001x₁ + 3x₂.
As x₁ increases by 1 unit, holding $x _ { 2 }$ constant, the value of y is estimated to decrease by 0.001units, on average.

Question 66

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.

Question 67

Given the multiple linear regression equation, ŷ = b₀ + b₁x₁ + b₂x₂, the value of b₂ is the estimated average increase in y for a one unit increase in x₂, whilst holding x₁ constant.

Question 68

In regression analysis, the total variation in the dependent variable y, measured by $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ , can be decomposed into two parts: the explained variation, measured by SSR, and the unexplained variation, measured by SSE.

Question 69

In a multiple regression analysis involving 50 observations and 5 independent variables, SST = 475 and SSE = 71.25. The multiple coefficient of determination is 0.85.

Question 70

A multiple regression analysis that includes 25 data points and 4 independent variables produces SST = 400 and SSR = 300. The multiple standard error of estimate will be 5.

Question 71

In testing the significance of a multiple regression model in which there are three independent variables, the null hypothesis is Ho: β₀ = β₁ = β₂ = β₃.

Question 72

In multiple regression, the standard error of estimate is defined by $S _ { \varepsilon } = \sqrt { SS E / ( n - k ) }$ , where n is the sample size and k is the number of independent variables.

Question 73

A multiple regression model has the form $\hat{y}$ = b₀ + b₁x₁ + b₂x₂. The coefficient b₂ is interpreted as the change in $y$ per unit change in x₂.

Question 74

Excel and Minitab print a second $R ^ { 2 }$ 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.

Question 75

A multiple regression the coefficient of determination is 0.81. The percentage of the variation in $y$ that is explained by the regression equation is 81%.

Question 76

For the multiple regression model $= 40 + 15 x _ { 1 } - 10 x _ { 2 } + 5 x _ { 3 }$ , if $x _ { 2 }$ were to increase by 5 units, holding $x _ { 1 }$ and $x _ { 3 }$ constant, the value of $y$ would decrease by 50 units, on average.

Question 77

In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the number of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are 4 and 20, respectively.

Question 78

The adjusted multiple coefficient of determination is adjusted for the number of independent variables and the sample size.

Question 79

In order to test the significance of a multiple regression model involving 4 independent variables and 30 observations, the number of degrees of freedom for the numerator and denominator for the critical value of F are 4 and 26, respectively.

Question 80

In a multiple regression, a large value of the test statistic F indicates that most of the variation in y is explained by the regression equation, and that the model is useful; while a small value of F indicates that most of the variation in y is unexplained by the regression equation, and that the model is useless.