Question 1

Which statistic is used to test a global hypothesis about a multiple regression equation?
A) t-statistic
B) z-statistic
C)  (chi-square statistic)
D) F

Accepted Answer

The F-statistic is used to test a global hypothesis about a multiple regression equation. It measures the overall significance of the regression model by comparing the explained variance to the residual variance. The higher the F-statistic, the more significant the regression model is. Therefore, the correct option is D, i.e., F.

Question 2

What can we conclude if the global test of regression rejects the null hypothesis?
A) Strong correlations exist among the variables.
B) No relationship exists between the dependent variable and any of the independent variables.
C) At least one of the net regression coefficients is not equal to zero.
D) Good predictions are not possible.

Accepted Answer

If the global test of regression rejects the null hypothesis, it means that at least one of the regression coefficients is not equal to zero, indicating that there is a significant relationship between the dependent variable and one or more independent variables. It does not necessarily mean that strong correlations exist or that good predictions are not possible. Option B is incorrect as it suggests that there is no relationship which is opposite to what rejecting the null hypothesis indicates.

Question 3

What can we conclude if the global test of regression does not reject the null hypothesis?
A) A strong relationship exists among the variables
B) No relationship exists between the dependent variable and any of the independent variables
C) The independent variables are good predictors
D) Good forecasts are possible

Accepted Answer

If the global test of regression does not reject the null hypothesis, it means that no significant relationship exists between the dependent variable and any of the independent variables. Therefore, option B is the correct choice. Options A, C, and D imply the presence of a significant relationship, which is not supported by the non-rejection of the null hypothesis.

Question 4

Multiple regression analysis is applied when analyzing the relationship between
A) An independent variable and several dependent variables
B) A dependent variable and several independent variables
C) Several dependent variables and several independent variables
D) Several regression equations and a single sample

Accepted Answer

Multiple regression analysis is used to analyze the relationship between a dependent variable and several independent variables. Therefore, the best choice is B, "A dependent variable and several independent variables."

Question 5

All other things being held constant, what is the change in the dependent variable for a unit change in the first independent variable for the multiple regression equation: Ŷ = 5.2 + 6.3X₁ - 7.1 X₂? A) -7.1 B) +6.3 C) +5.2 D) +4.4

Accepted Answer

The coefficient for the first independent variable (X₁) is 6.3. Therefore, for a unit change in X₁, the dependent variable Ŷ will change by 6.3 units, holding all other variables constant.

Question 6

In multiple regression analysis, a dummy variable is
A) An additional quantitative variable
B) A nominal variable with three or more values
C) A nominal variable with only two values
D) A new regression coefficient

Accepted Answer

A dummy variable is a nominal variable with only two values, often represented by 0 or 1, to indicate the presence or absence of a certain characteristic or category. It is used in regression analysis to capture the effect of categorical predictors on the dependent variable.

Question 7

The degrees of freedom associated with the regression sum of squares equals
A) the number of independent variables
B) 1
C) F-ratio
D) (n - 2)

Accepted Answer

The degrees of freedom for the regression sum of squares in a simple linear regression model is equal to the number of independent variables, which is 1.

Question 8

What does the correlation matrix for a multiple regression analysis contain?
A) Multiple correlation coefficients
B) Simple correlation coefficients
C) Multiple coefficients of determination
D) Multiple standard errors of estimate

Accepted Answer

The correlation matrix in a multiple regression analysis contains simple correlation coefficients, which measure the linear relationship between each pair of variables.

Question 9

In multiple regression analysis, residuals (Y - Ŷ) are used to:
A) Provide a global test of a multiple regression model.
B) Evaluate the assumption of linearity.
C) Calculate the variance inflation factor.
D) Compare two regression coefficients.

Accepted Answer

Residuals are used to evaluate the assumption of linearity in a multiple regression analysis. Residuals are the differences between the observed values and the predicted values (Y - Ŷ), and they indicate how well the model fits the data. The assumption of linearity requires that the relationship between the independent variables and the dependent variable is linear, and residuals can be used to check for nonlinearity. If the residuals show a pattern, such as a curve or a U shape, then nonlinearity may be present and the model may need to be revised.

Question 10

What happens as the scatter of data values about the regression plane increases? A) Standard error of estimate increases B) R² increases C) (1 - R²) decreases D) Error sum of squares decreases

Accepted Answer

As the scatter of data values about the regression plane increases, the variability in the data increases, resulting in a larger standard error of estimate.

Question 11

Which of the following is a characteristic of the F-distribution?
A) Normally distributed
B) Positively skewed
C) Negatively skewed
D) Equal to the t-distribution

Accepted Answer

The F-distribution is positively skewed, with most of the values concentrated on the left side and a long tail extending to the right.

Question 12

In multiple regression, a dummy variable is significantly related to the dependent variable when
A) the global test of the regression equation is rejected.
B) the test of the dummy variable's regression coefficient is rejected.
C) the dummy variable is correlated with other independent variables.
D) the dummy variable is coded as 2 or 3.

Accepted Answer

To determine if a dummy variable is significantly related to the dependent variable, we need to test the regression coefficient of the dummy variable using a t-test. If the null hypothesis (the coefficient equals zero) is rejected, then we can conclude that the dummy variable is significantly related to the dependent variable. The other options do not specifically address the significance of the dummy variable's impact on the dependent variable.

Question 13

Which statistic is used to test hypotheses about individual regression coefficients?
A) t-statistic
B) z-statistic
C)  (chi-square statistic)
D) F

Accepted Answer

The t-statistic is the appropriate statistic for testing hypotheses about individual regression coefficients. It measures the ratio of the estimated coefficient to its standard error, and follows a t-distribution under the assumption of normal errors.

Question 14

The coefficient of determination measures the proportion of
A) explained variation relative to total variation.
B) variation due to the relationship among variables.
C) error variation relative to total variation.
D) variation due to regression.

Accepted Answer

The coefficient of determination, also known as R-squared, measures the proportion of variation in the dependent variable that is explained by the independent variable(s). Therefore, it measures the proportion of explained variation relative to total variation.

Question 15

In a regression analysis, three independent variables are used in the equation based on a sample of forty observations. What are the degrees of freedom associated with the F-statistic?
A) 3 and 39
B) 4 and 40
C) 3 and 36
D) 2 and 39

Accepted Answer

The degrees of freedom associated with the F-test are calculated as follows: numerator degrees of freedom = number of independent variables in the model, which is 3 in this case; denominator degrees of freedom = sample size minus the number of independent variables, which is 40 - 3 = 37. Therefore, the correct choice is C, which shows the numerator degrees of freedom as 3 and denominator degrees of freedom as 36.

Question 16

The best example of a null hypothesis for testing an individual regression coefficient is: A) H₀: $\beta$₁ = $\beta$₂ = $\beta$₃ = $\beta$₄ = 0 B) H₀: µ₁ = µ₂ = µ₃ = µ₄ = 0 C) H₀: $\beta$₁ = 0 D) H₀: $\beta$₁ $\neq$ 0

Accepted Answer

The null hypothesis for testing an individual regression coefficient is that the coefficient is equal to zero, indicating no effect.

Question 17

In multiple regression analysis, residuals (Y - Ŷ) are used to:
A) Provide a global test of a multiple regression model.
B) Evaluate multicollinearity.
C) Evaluate homoscedasticity.
D) Compare two regression coefficients.

Accepted Answer

Residuals are used to evaluate homoscedasticity, which refers to the equal variance of errors across all levels of the independent variables. By examining the residuals, we can detect if the variance of errors is consistent across the range of independent variables.

Question 18

The best example of a null hypothesis for a global test of a multiple regression model is: A) H₀: $\beta$₁ = $\beta$₂ = $\beta$₃ = $\beta$₄ = 0 B) H₀: µ₁ = µ₂ = µ₃ = µ₄ = 0 C) H₀: $\beta$₁ = 0 D) If F is greater than 20.00 then reject.

Accepted Answer

The null hypothesis for a global test of a multiple regression model is that all the coefficients of the independent variables are equal to zero, indicating that none of the predictors have a significant effect on the dependent variable.

Question 19

In multiple regression analysis, residuals (Y - Ŷ) should be
A) qualitative variables.
B) significantly different from zero.
C) correlated.
D) normally distributed with a mean of zero.

Accepted Answer

Residuals should be normally distributed with a mean of zero in order for the regression analysis to be considered valid. Choice A is incorrect because residuals are quantitative, not qualitative. Choice B is incorrect because residuals being significantly different from zero would imply a lack of fit in the regression model. Choice C is incorrect because residuals should not be correlated with any of the independent variables.

Question 20

The best example of an alternate hypothesis for a global test of a multiple regression model is: A) H₁: $\beta$₁ = $\beta$₂ = $\beta$₃ = $\beta$₄ = 0 B) H₁: $\beta$₁ $\neq$$\beta$₂ $\neq$$\beta$₃ $\neq$$\beta$₄ $\neq$ C) H₁: Not all the $\beta$'s are equal to 0 D) If F is less than 20.00 then fail to reject.

Accepted Answer

The alternate hypothesis should state that not all the regression coefficients (F1F1F1S1F1F1F10's) are equal to zero. Option C clearly states this. Option A and B are both incorrect as they specify the specific values of the regression coefficients, which is not necessary for an alternate hypothesis. Option D is incorrect as it is a statement regarding the decision rule for hypothesis testing, rather than an actual alternate hypothesis.

Quiz 14: Multiple Regression Analysis

Which statistic is used to test a global hypothesis about a multiple regression equation?

What can we conclude if the global test of regression rejects the null hypothesis?

What can we conclude if the global test of regression does not reject the null hypothesis?

Multiple regression analysis is applied when analyzing the relationship between

All other things being held constant, what is the change in the dependent variable for a unit change in the first independent variable for the multiple regression equation: Ŷ = 5.2 + 6.3X1 - 7.1 X2?

In multiple regression analysis, a dummy variable is

The degrees of freedom associated with the regression sum of squares equals

What does the correlation matrix for a multiple regression analysis contain?

In multiple regression analysis, residuals (Y - Ŷ) are used to:

What happens as the scatter of data values about the regression plane increases?

Which of the following is a characteristic of the F-distribution?

In multiple regression, a dummy variable is significantly related to the dependent variable when

Which statistic is used to test hypotheses about individual regression coefficients?

The coefficient of determination measures the proportion of

In a regression analysis, three independent variables are used in the equation based on a sample of forty observations. What are the degrees of freedom associated with the F-statistic?

The best example of a null hypothesis for testing an individual regression coefficient is:

In multiple regression analysis, residuals (Y - Ŷ) are used to:

The best example of a null hypothesis for a global test of a multiple regression model is:

In multiple regression analysis, residuals (Y - Ŷ) should be

The best example of an alternate hypothesis for a global test of a multiple regression model is:

All other things being held constant, what is the change in the dependent variable for a unit change in the first independent variable for the multiple regression equation: Ŷ = 5.2 + 6.3X₁ - 7.1 X₂?