Deck 16: Multiple Regression

Question 1

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.

Answer 1

0.0.

Question 2

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.

Answer 2

multicollinearity.

Question 3

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.

Answer 3

All of these choices are correct.

Question 4

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 5

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 6

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 7

In a multiple regression analysis involving 30 data points, the standard error of estimate squared is calculated as s $\varepsilon$ 2 = 1.5 and the sum of squares for error as SSE = 36. The number of the independent variables must be:

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

Question 8

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.

Question 9

An estimated multiple regression model has the form ŷ = 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 10

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 11

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 12

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 13

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 14

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 15

The adjusted 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.

Question 16

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 coefficient of determination will be close to zero.
D)All of these choices are correct.

Question 17

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 18

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 coefficient of determination.

Question 19

In a multiple regression analysis involving 6 independent variables and a sample of 19 data points the total variation in y is SST = 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 20

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.

Question 21

An estimated multiple regression model has the form $\hat { y } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x _ { 1 } + \hat { \beta } _ { 2 } x _ { 2 } .$ Which of the following best describes $\hat { \beta } _ { 2 }$ ?

A)We estimate for each one unit increase in x2, that y will increase by $\hat { \beta } _ { 2 }$ units, on average.
B)Whilst holding x1 constant, we estimate for each one unit increase in x2, that y will increase by $\hat { \beta } _ { 2 }$ units, on average.
C)Whilst holding x2 constant, we estimate for each one unit increase in x1, that y will increase by $\hat { \beta } _ { 2 }$ units, on average
D)Whilst holding x1 constant, we estimate for each one unit increase in x2, that y will increase by $\hat { \beta } _ { 2 }$ units, on average .

Question 22

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 23

If none of the data points for a multiple regression model with two independent variables were on the regression plane, then the 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 24

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 25

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)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 coefficient of determination will assume a value close to zero.

Question 26

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 27

For a multiple regression model:

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

Question 28

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

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

Question 29

For the estimated multiple regression model ŷ = 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 30

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 31

In a multiple regression model, the following statistics are given: SSE = 100, R2 = 0.995, k = 5, n = 15. The 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 32

The coefficient of determination is defined as:

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

Question 33

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 34

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

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

Question 35

For a multiple regression model, the following statistics are given: Total variation in y = SST = 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 36

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

A)0 to 1
B)−1 to 1
C)1 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 37

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 38

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

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

Question 39

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 40

If all the points for a multiple regression model with two independent variables were on the regression plane, then the coefficient of determination would equal:

A) 0.
B) 1.
C) 2, since there are two independent variables.
D) any number between 0 and 2.

Question 41

For the multiple regression model $\hat { y }$ = 75 + 25x1 - 15 x2 + 10 x3, 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 42

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 coefficient of determination will be:

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

Question 43

In a regression model involving 60 observations, the following estimated regression model was obtained: $\hat { y }$ = 51.4 + 0.70x1 + 0.679 x2 - 0.378 x3. For this model, total variation in y = SST = 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 44

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 45

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 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

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 48

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 49

A multiple regression analysis that includes 20 data points and 4 independent variables results in total variation in y = SST = 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 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

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 52

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 53

In a regression model involving 30 observations, the following estimated regression model was obtained: $\hat { y }$ = 60 + 2.8x1 + 1.2 x2 - x3. For this model, total variation in y = SST = 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

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

Question 55

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

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

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 58

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

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

Question 59

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

Question 60

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 = SST = 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 61

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 62

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

Question 63

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 64

For the multiple regression model $\ddot { y } = 40 + 15 x _ { 1 } - 10 x _ { 2 } + 5 x _ { 3 }$ , 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 65

In reference to the equation $\hat { y }$ = 1.86 - 0.51x1 + 0.60 x2, the value 0.60 is the change in $y$ per unit change in $x _ { 2 }$ , regardless of the value of $x _ { 1 }$ .

Question 66

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 67

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 68

Given the multiple linear regression equation, $\hat { y } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x _ { 1 } + \hat { \beta } _ { 2 } x _ { 2 } ,$ the value of $\hat { \beta } _ { 2 }$ is the estimated average increase in y for a one unit increase in x2, whilst holding x1 constant.

Question 69

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 70

In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is $\hat { y }$ = 72 + 3.2x1 + 1.5 x2 - x3. 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 71

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

Question 72

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 73

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 74

Given the multiple linear regression equation, $\hat { y }$ = -0.80 + 0.12 x1 + 0.08 x2, the value -0.80 is the $y$ intercept.

Question 75

A multiple regression model has the form = 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 76

A multiple regression model has the form $\hat { y } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x _ { 1 } + \hat { \beta } _ { 2 } x _ { 2 } .$ The coefficient $\hat { \beta } _ { 2 }$ is interpreted as the change in $y$ per unit change in x2.

Question 77

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 78

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

Question 79

Excel prints 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 80

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%.