Question 1

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

Accepted Answer

Multiple regression involves the use of more than one independent variable to predict or explain the variation in a dependent variable.

Question 2

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.

Accepted Answer

This is a true statement. Each $\beta$ represents the change in the dependent variable associated with a one-unit change in the corresponding independent variable, while holding all other variables constant. This is known as the partial effect or partial regression coefficient.

Question 3

In a regression model involving 60 observations, the following estimated regression model was obtained: $= 51.4 + 0.70 x _ { 1 } + 0.679 x _ { 2 } - 0.378 x _ { 3 }$ For this model, total variation in y = SSY = 119,724 and SSR = 29,029.72. The value of MSE is:
A) 1619.541.
B) 9676.572.
C) 1995.400.
D) 5020.235.

Accepted Answer

The answer of In a regression model involving 60 observations,...

Question 4

In a regression model involving 30 observations, the following estimated regression model was obtained: $ = 60 + 2.8 x _ { 1 } + 1.2 x _ { 2 } - x _ { 3 }$ . For this model, total variation in y = SSY = 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.

Accepted Answer

The answer of In a regression model involving 30 observations,...

Question 5

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

Accepted Answer

Multiple linear regression involves using more than one independent variable to predict the dependent variable.

Question 6

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

Accepted Answer

Multicollinearity refers to a situation in which the independent variables are highly correlated with each other, not with the dependent variable. This can cause issues in regression analysis as it can be difficult to determine the unique effects of each independent variable on the dependent variable.

Question 7

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.

Accepted Answer

The degrees of freedom for t-tests in multiple regression analysis are calculated as n-k-1, where n is the number of observations and k is the number of independent variables. Therefore, in this case, the degrees of freedom would be 110-9-1=100.

Question 8

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.

Accepted Answer

MSE = SSE/(n-k-1) where n is the number of observations and k is the number of independent variables in the model. 
In this case, n = 50, k = 4 (including the intercept term). 
So, MSE = 175/(50-4-1) = 3.804. 
Therefore, the correct answer is D.

Question 9

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of $b _ { 2 }$ , this represents the probability that:
A) $\left| b _ { 2 } \right|$ could be this large if Ho: $\beta _ { 2 } \neq 0$ is really true.
B) $\left| b _ { 2 } \right|$ could be this large if Ho: $\beta _ { 2 } = 0$ is really false.
C) $\left| b _ { 2 } \right|$ could be this large if Ho: $\beta _ { 2 } \neq 0$ is really true
D) $\left| b _ { 2 } \right|$ could be this large if Ho: $\beta _ { 2 } = 0$ is really true..

Accepted Answer

The p-value for a coefficient in a regression model tests the null hypothesis that the true coefficient ($\beta_2$) is equal to zero. It represents the probability of observing a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true.

Question 10

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.

Accepted Answer

First-order autocorrelation is a condition in which a relationship exists between consecutive residuals. This means that the error term in a regression model is correlated with the error term in the immediately preceding observation. This violates the assumption of independence of residuals and can lead to biased or inefficient estimates of the regression coefficients.

Question 11

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.

Accepted Answer

The formula for the multiple standard error of estimate (S[subscript]e) is:

S[subscript]e = sqrt (SSE / (n - k - 1))

where SSE is the sum of the squared differences between observed and predicted values of y, n is the sample size, and k is the number of independent variables.

Substituting the given values, we get:

S[subscript]e = sqrt (180 / (20 - 4 - 1)) = 3.464

Therefore, the multiple standard error of estimate will be 3.464, which can be found in choice B.

Question 12

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 multiple coefficient of determination will be:
A) 0.667.
B) 0.600.
C) 0.400.
D) 0.200.

Accepted Answer

The answer of A multiple regression analysis that includes 4...

Question 13

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.

Accepted Answer

The Durbin-Watson test is used to determine if there is first-order autocorrelation. It checks for the presence of correlation between residuals from one observation to the next in a time series or panel data regression model.

Question 14

A multiple regression analysis that includes 20 data points and 4 independent variables results in total variation in y = SSY = 200 and SSR = 160. The multiple standard error of estimate will be:
A) 0.80.
B) 3.266.
C) 3.651.
D) 1.633.

Accepted Answer

The multiple standard error of estimate is calculated using the formula: SEE = sqrt(SSE / (n - k - 1)), where SSE = SSY - SSR. Here, SSE = 200 - 160 = 40, n = 20, and k = 4. Thus, SEE = sqrt(40 / (20 - 4 - 1)) = sqrt(40 / 15) = 1.633.

Question 15

For the multiple regression model $ = 75 + 25 x _ { 1 } - 15 x _ { 2 } + 10 x _ { 3 }$ , 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.

Accepted Answer

The answer of For the multiple regression model \( =...

Question 16

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) $H _ { 0 } : \beta _ { 0 } = \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } \neq 0$ .

Accepted Answer

The null hypothesis for testing the overall significance of a multiple regression model is that all of the regression coefficients (betas) are equal to zero, which means that none of the independent variables are related to the dependent variable. Therefore, the correct choice is C, which states that all the betas are equal to zero.

Question 17

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%

Accepted Answer

The coefficient of determination (R-squared) is the proportion of the variance in the dependent variable (y) that is explained by the independent variables in the regression equation. In this case, the R-squared value is 0.64, which means that 64% of the variation in y can be explained by the regression model. Therefore, the answer is D - 64%.

Question 18

Given the multiple linear regression equation  $= - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value -0.80 is the $y$ intercept.

Accepted Answer

The answer of Given the multiple linear regression equation ...

Question 19

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 = SSY = 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.

Accepted Answer

The F-value of the test statistic is calculated as:

F = (SSR / k) / (SSE / (n - k - 1))

where k is the number of independent variables and n is the number of data points.

In this case, k = 5 and n = 25. Plugging in the given values, we get:

F = (300 / 5) / (200 / 19) = 15 / 10 = 1.5

Therefore, the F-value of the test statistic is 1.5.

Comparing this value to the F-distribution table with degrees of freedom (k, n - k - 1) = (5, 19) at α = 0.05, we find that the critical F-value is 2.45. Since 1.5 < 2.45, we fail to reject the null hypothesis that all of the regression coefficients are equal to 0. Therefore, the regression model is not significant.

Question 20

For a set of 30 data points, Excel has found the estimated multiple regression equation to be = -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.

Accepted Answer

The answer of For a set of 30 data points,...

Quiz 19: Multiple Regression

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

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.

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

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

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:

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:

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of $b _ { 2 }$ , this represents the probability that:

Which of the following best describes first-order autocorrelation?

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 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 multiple coefficient of determination will be:

Which of the following best describes the Durbin-Watson test?

A multiple regression analysis that includes 20 data points and 4 independent variables results in total variation in y = SSY = 200 and SSR = 160. The multiple standard error of estimate will be:

In testing the validity of a multiple regression model in which there are four independent variables, the null hypothesis is:

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:

Given the multiple linear regression equation $Given the multiple linear regression equation = - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 } , the value -0.80 is the y intercept.$ $= - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value -0.80 is the $y$ intercept.

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 = SSY = 500, SSR = 300, and SSE = 200. In testing the validity of the regression model, the F-value of the test statistic will be:

Quiz 19: Multiple Regression

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

For each x term in the multiple regression equation, the corresponding β\betaβ is referred to as a partial regression coefficient or slope of the independent variable.

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

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

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:

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:

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of b2b _ { 2 }b2​ , this represents the probability that:

Which of the following best describes first-order autocorrelation?

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 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 multiple coefficient of determination will be:

Which of the following best describes the Durbin-Watson test?

A multiple regression analysis that includes 20 data points and 4 independent variables results in total variation in y = SSY = 200 and SSR = 160. The multiple standard error of estimate will be:

For the multiple regression model =75+25x1−15x2+10x3 = 75 + 25 x _ { 1 } - 15 x _ { 2 } + 10 x _ { 3 }=75+25x1​−15x2​+10x3​ , if x2x _ { 2 }x2​ were to increase by 5, holding x1x _ { 1 }x1​ and X3X _ { 3 }X3​ constant, the value of y would:

In testing the validity of a multiple regression model in which there are four independent variables, the null hypothesis is:

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:

Given the multiple linear regression equation =−0.80+0.12x1+0.08x2= - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }=−0.80+0.12x1​+0.08x2​ , the value -0.80 is the yyy intercept.

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 = SSY = 500, SSR = 300, and SSE = 200. In testing the validity of the regression model, the F-value of the test statistic will be:

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.

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:

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of $b _ { 2 }$ , this represents the probability that:

Given the multiple linear regression equation $Given the multiple linear regression equation = - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 } , the value -0.80 is the y intercept.$ $= - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value -0.80 is the $y$ intercept.