Question 1

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.
D) It is computed as the ratio of total variation to explained variation and in the case of two independent variables, it describes how well the regression hyperplane fits the data.
E) None of these.

Accepted Answer

The coefficient of multiple determination (R-squared) is denoted by and represents the proportion of the variance in the dependent variable (y) that is explained by the independent variables (predictors) in the regression model. Choice B is incorrect as it falsely implies that the ratio of total variation to explained variation is the coefficient of determination, which is the square of the correlation coefficient. Choice C is incorrect as it is only applicable to two independent variables and disregards the other cases. Choice D combines the incorrect statement from option B with the limitation of option C.

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
E) number of independent variables and the significance level

Accepted Answer

The adjusted multiple coefficient of determination accounts for the number of independent variables and the sample size in the regression model. It adjusts for the potential overfitting that can occur with the multiple coefficient of determination when too many independent variables are included in the model relative to the sample size. It does not take into account the y-intercept, number of dependent variables, coefficient of correlation, or significance level.

Question 3

A three-variable multiple regression establishes an estimated multiple regression equation in such a way that:
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 plan, 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
D) all of these
E) none of these

Accepted Answer

The answer of A three-variable multiple regression establishes an estimated...

Question 4

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. Then, the value of MSR is:
A) 12.525
B) 33
C) 167
D) 15
E) 18.52

Accepted Answer

MSR = SSR/df(regression) = SSR/k = 501/3 = 167 (where k is the number of predictors in the model)
Therefore, the answer is choice C.

Question 5

For a multiple regression model the following statistics are given: Total SS = 400, SSR = 350, k = 4, and n = 20. The coefficient of determination adjusted for degrees of freedom is:
A) 87.5%
B) 15.83%
C) 12.5%
D) 84.17%
E) 16.54%

Accepted Answer

The coefficient of determination (R-squared) is given by the ratio SSR/Total SS.

So, R-squared = SSR/Total SS = 350/400 = 0.875

The adjusted R-squared formula takes into account the number of predictors and the sample size:

Adjusted R-squared = 1 - [(1 - R-squared)(n - 1)/(n - k - 1)]

Plugging in the numbers:

Adjusted R-squared = 1 - [(1 - 0.875)(20 - 1)/(20 - 4 - 1)] = 0.8417 or 84.17%

Therefore, the answer is D.

Question 6

Given MSR = 345 and MSE = 431.25, the value of the F-statistic:
A) equals .894
B) equals 1.25
C) equals .80
D) equals .43
E) cannot be computed without additional information

Accepted Answer

The F-statistic is calculated by dividing MSR by MSE. Therefore, F = MSR/MSE = 345/431.25 ≈ 0.80. Thus, the answer is choice C.

Question 7

In testing the validity of a multiple regression model, a large value of the F-test statistic indicates that:
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
E) most of the variation in y is explained by the regression equation and the model provides a poor fit

Accepted Answer

A large value of the F-test statistic in the context of a multiple regression model suggests that the regression model explains a significant portion of the variance in the dependent variable (y), indicating a good fit of the model to the data.

Question 8

In a multiple regression model, the standard deviation of the error variable  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) 0
E) Not enough information is given to answer this question.

Accepted Answer

In a multiple regression model, the standard deviation of the error variable (also known as the residual) is assumed to be constant for all values of the independent variables. This assumption is known as homoscedasticity. If the standard deviation of the residuals is not constant across the range of the independent variables, it is known as heteroscedasticity, which violates one of the assumptions of the multiple regression model.

Question 9

A multiple regression model has the form:  . As  increases by 1 unit, with  and  held constant, then y on average is expected to:
A) increase by 1 unit
B) increase by 10 units
C) decrease by 3 units
D) decrease by 4 units
E) decrease by 5 units

Accepted Answer

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

Question 10

For a multiple regression model, the following statistics are given: Total SS = 500, SSE = 60, and n = 20. The coefficient of determination, expressed as a percentage, is:
A) 88%
B) 12%
C) 93.81%
D) 77.44%
E) 7.51%

Accepted Answer

The coefficient of determination, $R^2$, is calculated as $1 - \frac{SSE}{TSS}$, where SSE is the sum of squares due to error, and TSS is the total sum of squares. Given $TSS = 500$ and $SSE = 60$, $R^2 = 1 - \frac{60}{500} = 1 - 0.12 = 0.88$, or 88% when expressed as a percentage.

Question 11

A multiple regression model involves 5 predictor variables and 30 observations. If we want to test the hypotheses  at the 5% significance level, the critical value of the rejection region will be:
A) 1.711
B) 2.621
C) 2.045
D) 1.699
E) 1.341

Accepted Answer

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

Question 12

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, the critical value of the rejection region is:
A) 5.93
B) 7.09
C) 5.29
D) 6.30
E) 4,54

Accepted Answer

The answer of A multiple regression model involves 3 predictor...

Question 13

Which of the following statements is false?
A) An adjusted coefficient of multiple determination, denoted by(adj.), is adjusted for the degrees of freedom.
B) 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.
C) The coefficient of multiple determination takes on values between 0 and 1, inclusive.
D) 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).
E) None of these statements is false.

Accepted Answer

The coefficient of multiple correlation, denoted by R, is actually the square root of the coefficient of multiple determination (R-squared), and it does not directly represent the proportion of total variation explained by the regression.

Question 14

Which of the following statements is correct regarding 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 coefficient a represents the slope of the line.
C) The estimated regression coefficientequals the change in estimated Y that is associated with a unit change inand.
D) All are correct.
E) None of these are correct.

Accepted Answer

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

Question 15

Given a multiple regression with a regression sum of squares of 850 and a total sum of square of 1,000, the coefficient of multiple determination:
A) equals .150
B) equals .850
C) equals .387
D) equals .459
E) cannot be determined without additional information

Accepted Answer

The coefficient of determination, R², is the proportion of the total variability in the data that is explained by the regression model. It is calculated as the ratio of the regression sum of squares (RSS) to the total sum of squares (TSS):

R² = RSS / TSS

In this case, R² = 850 / 1000 = 0.85, which means that the regression model explains 85% of the total variability in the data. Therefore, the best choice is B) equals .850.

Question 16

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 zero and that, therefore, all of 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) It is a test of the significance of individual coefficients, rather than of the overall significance of the regression model and it is a test of the overall significance of a regression, rather than a test of the significance of individual coefficients.
E) None of these.

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. This test helps to determine whether the overall model is significant in predicting the dependent variable. It does not focus on the individual significance of each predictor variable.

Question 17

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) It does not affect the F-test of the analysis of variance.
E) All of these.

Accepted Answer

Multicollinearity is a condition that exists when the independent variables are highly correlated with each other, not with the dependent variable. Therefore, statement C is not true.

Question 18

Which of the following statements is correct?
A) Simple regression analysis helps us predict the value of one variable from the known value of another variable.
B) Simple correlation analysis helps us measure the overall strength of association between two variables.
C) Multiple correlation analysis measures the overall strength of association among more than two variables.
D) All of these.
E) None of these.

Accepted Answer

A) Simple regression analysis does indeed help us predict the value of one variable from the known value of another variable.

B) Simple correlation analysis does indeed help us measure the overall strength of association between two variables.

C) Multiple correlation analysis does indeed measure the overall strength of association among more than two variables.

Therefore, all three statements are correct, making choice D the best answer.

Question 19

In multiple regression analysis, the ratio MSR/MSE yields the:
A) multiple coefficient of determination
B) F-test statistic for testing the validity of the regression equation
C) t-test statistic for testing each individual regression coefficient
D) adjusted multiple coefficient of determination
E) F-test statistic for testing the validity of the regression equation and t-test statistic for testing each individual regression coefficient

Accepted Answer

The answer of In multiple regression analysis, the ratio MSR/MSE...

Question 20

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 ration of an estimated partial-regression coefficient to its standard error.
E) All of these.

Accepted Answer

The p-value measures the probability of observing the data, or something more extreme, if the null hypothesis is true. It does not directly tell us the probability that the null hypothesis is true itself, but rather indicates how compatible our data is with the null hypothesis. Choices A, C, and D describe other statistical concepts: A refers to the interpretation of a regression coefficient, C describes a measure related to multiple correlation, and D describes the calculation of a t-statistic, not a p-value.

Quiz 13: Multiple Regression Analysis

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

The adjusted multiple coefficient of determination is adjusted for the:

A three-variable multiple regression establishes an estimated multiple regression equation in such a way that:

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. Then, the value of MSR is:

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

Given MSR = 345 and MSE = 431.25, the value of the F-statistic:

In testing the validity of a multiple regression model, a large value of the F-test statistic indicates that:

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

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

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

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

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, the critical value of the rejection region is:

Which of the following statements is false?

Which of the following statements is correct regarding the multiple-regression equation: ?

Given a multiple regression with a regression sum of squares of 850 and a total sum of square of 1,000, the coefficient of multiple determination:

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

Which of the following statements regarding multicollinearity is not true?

Which of the following statements is correct?

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

Which of the following correctly describes a p-value?