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

Accepted Answer

MSR (Mean Square Regression) = SSR/(number of predictors) = SSR/(1) = 501.

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.

Accepted Answer

A p-value is the probability of observing the data, or something more extreme, assuming the null hypothesis is true.

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

Accepted Answer

The F-test statistic is calculated as the ratio of the mean square for regression (MSR) to the mean square for error (MSE). First, calculate the regression sum of squares (SSR) as 800 - 200 = 600. Then, MSR = SSR/4 = 150 and MSE = 200/(25-4-1) = 10. The F-statistic is MSR/MSE = 150/10 = 15.

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)

Accepted Answer

The answer of To test the validity of a multiple...

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%

Accepted Answer

The coefficient of determination, also known as R-squared, is the proportion of variance in the dependent variable that is explained by the independent variables. It is calculated as (Total SS - SSE) / Total SS. Plugging in the given values, we get R-squared = (500 - 60) / 500 = 0.88 or 88%. Therefore, the best choice is B.

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.

Accepted Answer

The coefficient of multiple determination (R-squared) is denoted by R-sq and represents the proportion of the total variation in the dependent variable (y) that is explained by the regression model with one or more predictor variables. It ranges from 0 to 1, where R-sq=0 indicates that the model explains none of the variation, whereas R-sq=1 indicates that the model explains all of the variation in the dependent variable. Therefore, choice A correctly describes the coefficient of multiple determination. Choice B describes the F-statistic, which is used to test the overall significance of the regression model. Choice C describes the coefficient of determination in the case of two independent variables, which is equivalent to R-sq.

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

Accepted Answer

The ratio MSR/MSE (mean square regression/mean square error) is used to calculate the F-test statistic for testing the overall significance of the regression model.

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

Accepted Answer

The answer of Suppose a regression analysis based on the...

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%

Accepted Answer

The coefficient of determination is given by R² = SSR/SS. So, R² = 350/400 = 0.875 or 87.5%.

To adjust for degrees of freedom, we use the formula R_adj² = 1 - ((n-1)/(n-k-1))(1-R²).

Plugging in the given values, we get:
R_adj² = 1 - ((20-1)/(20-4-1))(1-0.875) = 0.8417 or 84.17%

Therefore, the best choice is B.

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.

Accepted Answer

In a three-variable multiple regression, the estimates fall on a regression plane, which minimizes the sum of squared vertical deviations, not horizontal deviations.

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

Accepted Answer

The answer of In order to test the validity of...

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)

Accepted Answer

The answer of A three-variable multiple regression plane is positioned...

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.

Accepted Answer

The answer of A multiple regression model has the form...

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

Accepted Answer

The F test statistic for testing the usefulness of the regression model is found by taking the ratio of the mean square regression (MSR) to the mean square error (MSE). This tests whether the amount of variability explained by the regression model is significantly larger than the amount of variability not explained by the model. Choice A is the expression for this ratio.

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.

Accepted Answer

The answer of Which of the following is the correct...

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

Accepted Answer

The answer of A multiple regression model involves 5 predictor...

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

Accepted Answer

The adjusted multiple coefficient of determination is adjusted for the number of predictor variables and the sample size. It takes into account the possibility that adding more predictor variables may improve the overall fit of the model, but may not necessarily improve the predictability of the dependent variable. It penalizes complex models to avoid overfitting, which can lead to poor generalization of the model to new data.

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) 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 B

Accepted Answer

A multiple-regression ANOVA test is a test of the overall significance of a regression, rather than a test of the significance of individual coefficients. It tests whether the regression equation as a whole is statistically significant and explains a significant amount of the variation in the dependent variable. Individual coefficients can be tested separately using t-tests, but the multiple-regression ANOVA test is focused on the overall model's significance.

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

Accepted Answer

The critical value of the rejection region for a multiple regression model with 3 predictor variables and a sample of 20 data points at the 1% significance level is 5.29. This value is obtained from the F-distribution with 3 and 16 degrees of freedom.

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

Accepted Answer

The answer of In a multiple regression analysis involving 24...

Quiz 13: Multiple Regression Analysis

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?

Which of the following correctly describes a p-value?

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?

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?

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?

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

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

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 ?

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

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

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?

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

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?

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

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

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?

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?