Question 1

In a multiple regression model, what is assumed to be true of the standard deviation of the error variable  ?
A) It is assumed to be constant for all values of the independent variables.
B) It is assumed to be constant for all values of the dependent variable.
C) It is assumed to be equal to 1.0.
D) It is assumed to be less than 1.0.

Accepted Answer

In multiple regression, it is assumed that the standard deviation of the error variable (also known as the residual) is constant (homoscedastic) for all values of the independent variables. This is because the model assumes that the relationship between the dependent variable and the independent variables is linear and that the error term is normally distributed with a constant variance. In other words, the spread of the residuals around the regression line should be roughly the same across all levels of the independent variables, indicating that the variability of the dependent variable around its mean is the same for all levels of the independent variables.

Question 2

In multiple regression models, which of the following may be assumed in with respect to the values of the error variable  ?
A) They are autocorrelated.
B) They are dependent on each other.
C) They are independent of each other.
D) They are always positive.

Accepted Answer

In multiple regression models, it is assumed that the errors are independent of each other. If they were dependent on each other, it would indicate a violation of the independence assumption and could lead to biased estimates. Autocorrelation and strictly positive errors are possible but not always assumed.

Question 3

A multiple regression model has the form  . Which of the following is the best interpretation of the coefficient  ?
A) It is the change in y per unit change in.
B) It is the change in y per unit change in, holdingconstant.
C) It is the change in y per unit change in, whenandvalues are correlated.
D) It is the change in the average value of y per unit change in, holdingconstant.

Accepted Answer

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

Question 4

In testing the validity of a multiple regression model involving 10 independent variables and 100 observations, what will the respective values of the numerator and denominator degrees of freedom for the critical value of F be?
A) 9 and 10
B) 9 and 90
C) 10 and 89
D) 10 and 100

Accepted Answer

The numerator degrees of freedom will be the number of independent variables, which is 10. The denominator degrees of freedom will be the difference between the total number of observations (100) and the number of independent variables (10), which is 90. Therefore, the critical value of F will have 10 and 89 degrees of freedom, which corresponds to choice C.

Question 5

In testing the validity of a multiple regression model, a large value of the F-test statistic is indicative of which of the following situations?
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.

Accepted Answer

A large F-test statistic suggests that the regression model explains a significant portion of the variation in the dependent variable, y, indicating a good fit.

Question 6

To test the validity of a multiple regression model, which of the following tests do we use for the null hypothesis that the regression coefficients are all 0?
A) t test
B) z test
C) F test
D) Rtest

Accepted Answer

The F test is used to test the null hypothesis that all the regression coefficients are equal to 0. It measures the overall significance of the regression model by comparing the variation explained by the regression to the remaining unexplained variation. The t test and z test are used to test the significance of individual regression coefficients, not the overall model. The R test is not a commonly used statistical test.

Question 7

Given MSR = 345 and MSE = 431.25, what is the value of the F statistic?
A) 1.25
B) 0.994
C) 0.894
D) 0.800

Accepted Answer

The answer of Given MSR = 345 and MSE =...

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. What is the adjusted multiple coefficient of determination?
A) 0.385
B) 0.591
C) 0.736
D) 0.877

Accepted Answer

The adjusted R-squared accounts for the number of predictors in the model and the sample size. It is calculated using the formula: Adjusted R² = 1 - [(1-R²)(n-1)/(n-k-1)], where R² is the unadjusted R-squared, n is the number of observations, and k is the number of predictors. Plugging in the values: Adjusted R² = 1 - [(1-0.769)(25-1)/(25-3-1)] = 0.736.

Question 9

A multiple regression model involves five independent variables and a sample of ten data points. If we want to test the validity of the model at the 5% significance level, what is the critical value?
A) 9.36
B) 6.26
C) 4.24
D) 3.33

Accepted Answer

The answer of A multiple regression model involves five independent...

Question 10

In a simple linear regression , the following pairs of (  ) are given: (6.75, 7.42), (8.96, 8.06), (10.30, 11.65), and (13.24, 12.15). Which of these values is equal to the sum of squares for error?
A) -0.0300
B) 4.2695
C) 39.2500
D) 39.2800

Accepted Answer

The answer of In a simple linear regression , the...

Question 11

What is the range in values of the coefficient of multiple determination?
A) from 1.0 to
B) from 0.0 to 1.0
C) from 1.0 to k, where k is the number of independent variables in the model
D) from 1.0 to n, where n is the number of observations in the dependent variable

Accepted Answer

The coefficient of multiple determination, or R-squared, measures the proportion of variance in the dependent variable that is explained by the independent variables in the model. R-squared values range from 0.0 to 1.0, where 0.0 indicates no relationship between the variables and 1.0 indicates a perfect relationship, meaning all of the variance in the dependent variable can be explained by the independent variables. Therefore, the best choice is B because it correctly identifies the range of values for R-squared.

Question 12

In a multiple regression model, if the residuals have a constant variance, which of the following should be evident?
A) The residual mean should be close to 0, and the residual standard deviation should be close to 1.0.
B) The plot of residuals against each independent variable should show that the spread in the residuals is about the same at all levels of the independent variables.
C) The residuals should have a variance equal to 0 for all levels of the independent variable.

Accepted Answer

If the residuals have a constant variance, it means that there is no heteroscedasticity in the model. In other words, the spread of the residuals should be about the same at all levels of the independent variables. This can be checked by plotting the residuals against each independent variable. Option A is not necessarily true, as the mean and standard deviation of the residuals can vary depending on the characteristics of the data. Option C is not true, as the residuals should have a non-zero variance.

Question 13

To test the validity of a multiple regression model involving two independent variables, which of the following is the most appropriate null hypothesis?
A)
B)
C)
D)

Accepted Answer

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

Question 14

In multiple regression analysis, which of the following is equal to the ratio MSR/MSE?
A) the t-test statistic for testing each individual regression coefficient
B) the F-test statistic for testing the validity of the regression equation
C) the multiple coefficient of determination
D) the adjusted multiple coefficient of determination

Accepted Answer

MSR (Mean Square Regression) is the sum of squares regression divided by the degrees of freedom for regression, and MSE (Mean Square Error) is the sum of squares error divided by the degrees of freedom for error. The ratio of MSR/MSE is the F-test statistic for testing the validity of the regression equation.

Question 15

In order to test the validity of a multiple regression model involving 5 independent variables and 30 observations, what are the respective numerator and denominator degrees of freedom for the critical value of F?
A) 5 and 24
B) 5 and 30
C) 6 and 25
D) 6 and 29

Accepted Answer

The numerator degrees of freedom is the number of independent variables, which is 5. The denominator degrees of freedom is the total number of observations minus the number of parameters estimated (including the intercept), which is 30 - 6 = 24.

Question 16

Given a multiple regression with a regression sum of squares of 850 and a total sum of square of 1000, what is the coefficient of multiple determination?
A) 0.150
B) 0.387
C) 0.850
D) 0.925

Accepted Answer

The coefficient of multiple determination, R², is calculated as the regression sum of squares divided by the total sum of squares. Here, R² = 850 / 1000 = 0.850.

Question 17

A multiple regression equation includes five independent variables, and the coefficient of determination is 0.81. What is the percentage of the variation in y that is explained by the regression equation?
A) 90%
B) 86%
C) 81%
D) about 16%

Accepted Answer

The coefficient of determination (R-squared) represents the proportion of variation in the dependent variable that is explained by the independent variables in the regression equation. The value of R-squared ranges between 0 and 1, where 0 indicates no linear relationship between the dependent and independent variables, and 1 indicates a perfect linear relationship.

In this case, the R-squared is 0.81, which means that the independent variables in the regression equation explain 81% of the variation in the dependent variable. Therefore, the correct answer is C, which represents 81%.

Question 18

Which of the following best describes the model  ?
A) It is a third-order polynomial model.
B) It is a second-order polynomial model.
C) It is a first-order polynomial model since there is only one independent variable x.
D) It is a quadratic model since the termwill often be dropped out of the model based on sample information.

Accepted Answer

The answer of Which of the following best describes the...

Question 19

For which of the following quantities is the adjusted multiple coefficient of determination adjusted??
A) the number of regression parameters including the y-intercept
B) the number of dependent variables and the sample size
C) the number of independent variables and the sample size
D) the coefficient of correlation and the significance level

Accepted Answer

The adjusted multiple coefficient of determination is adjusted for the number of independent variables and the sample size. This adjustment accounts for the possibility that adding more independent variables to the model may increase the R-squared even if they are not statistically significant, which can lead to overfitting.

Question 20

In multiple regression analysis involving 10 independent variables and 100 observations, how many degrees of freedom will the critical value of t for testing individual coefficients in the model have?
A) 100 degrees of freedom
B) 89 degrees of freedom
C) 10 degrees of freedom
D) 9 degrees of freedom

Accepted Answer

The degrees of freedom for the t-test in multiple regression is calculated as the number of observations minus the number of parameters estimated (including the intercept). Here, it is 100 - (10 + 1) = 89 degrees of freedom.

Quiz 13: Multiple Regression Analysis

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

In multiple regression models, which of the following may be assumed in with respect to the values of the error variable ?

A multiple regression model has the form . Which of the following is the best interpretation of the coefficient ?

In testing the validity of a multiple regression model involving 10 independent variables and 100 observations, what will the respective values of the numerator and denominator degrees of freedom for the critical value of F be?

In testing the validity of a multiple regression model, a large value of the F-test statistic is indicative of which of the following situations?

To test the validity of a multiple regression model, which of the following tests do we use for the null hypothesis that the regression coefficients are all 0?

Given MSR = 345 and MSE = 431.25, what is the value of the F statistic?

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. What is the adjusted multiple coefficient of determination?

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

In a simple linear regression , the following pairs of ( ) are given: (6.75, 7.42) , (8.96, 8.06) , (10.30, 11.65) , and (13.24, 12.15) . Which of these values is equal to the sum of squares for error?

What is the range in values of the coefficient of multiple determination?

In a multiple regression model, if the residuals have a constant variance, which of the following should be evident?

To test the validity of a multiple regression model involving two independent variables, which of the following is the most appropriate null hypothesis?

In multiple regression analysis, which of the following is equal to the ratio MSR/MSE?

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

Given a multiple regression with a regression sum of squares of 850 and a total sum of square of 1000, what is the coefficient of multiple determination?

A multiple regression equation includes five independent variables, and the coefficient of determination is 0.81. What is the percentage of the variation in y that is explained by the regression equation?

Which of the following best describes the model ?

For which of the following quantities is the adjusted multiple coefficient of determination adjusted??

In multiple regression analysis involving 10 independent variables and 100 observations, how many degrees of freedom will the critical value of t for testing individual coefficients in the model have?