Question 1

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 represents the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis that $\beta _ { 2 } = 0$ is true.

Question 2

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 3

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 4

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

In multiple regression, each $\beta$ coefficient represents the change in the dependent variable for a one-unit change in the corresponding independent variable, holding all other variables constant.

Question 5

For a set of 30 data points, Excel has found the estimated multiple regression equation to be $\hat{y}$ = -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,...

Question 6

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) measures the percentage of variation 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, meaning that 64% of the variation in y is explained by the regression equation. Therefore, the correct answer is D, 64%.

Question 7

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 answer of A multiple regression analysis that includes 20...

Question 8

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 9

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 10

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 in a multiple regression model typically tests whether all the coefficients of the independent variables are equal to zero, indicating that they have no effect on the dependent variable.

Question 11

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

Accepted Answer

In multiple regression, there are two or more independent variables that are used to predict the dependent variable. Therefore, the descriptor 'multiple' refers to having multiple independent variables.

Question 12

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

To find the MSE, we use the formula:

MSE = SSE / (n - k)

where n is the number of observations and k is the number of independent variables in the model (not counting the intercept).

In this case, n = 50 and k = 4 (there are four independent variables in the model).

Plugging in the values, we get:

MSE = 175 / (50 - 4) = 3.804

Therefore, the answer is D.

Question 13

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 14

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

To calculate the F-value, we use the formula F = (SSR / p) / (SSE / n - p - 1), where p is the number of independent variables and n is the number of data points. Plugging in the given values, we get F = (300/5) / (200/19) = 5.70, which corresponds to choice A.

Question 15

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 critical value of t in multiple regression analysis is determined by the degrees of freedom, which is calculated as n-k-1, where n is the sample size (110 observations) and k is the number of independent variables (9). Therefore, the degrees of freedom in this scenario would be 100 (110-9-1), making the correct answer D.

Question 16

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 a statistical test for the presence of first-order autocorrelation in the residuals (errors) of a regression analysis. Therefore, option C is the correct choice.

Question 17

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 residual errors in a time series are correlated with the errors in the previous time period.

Question 18

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 to calculate the multiple standard error of estimate (MSSE) is:

MSSE = sqrt (SSE / (n - p - 1))

where SSE is the sum of squared errors, n is the sample size and p is the number of independent variables.

In this case, SSE = 180, n = 20, and p = 4.

MSSE = sqrt (180 / (20 - 4 - 1)) = sqrt (180 / 15) = sqrt (12) = 3.464

Therefore, the MSSE is 3.464, which corresponds to answer choice B.

Question 19

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

Accepted Answer

This is the definition of multiple linear regression.

Question 20

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

Accepted Answer

Multicollinearity is a situation in which the independent variables are highly correlated with each other, not necessarily with the dependent variable. This can cause problems in statistical analysis, including inflated standard errors and difficulty in determining the true independent effect of each variable on the dependent variable.

Quiz 19: Multiple Regression

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.

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.

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 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 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:

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

In multiple regression, the descriptor 'multiple' refers to more than one 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:

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:

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:

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

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:

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.

Quiz 19: Multiple Regression

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:

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:

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.

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.

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 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 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:

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

In multiple regression, the descriptor 'multiple' refers to more than one 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:

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:

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:

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

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:

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.

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.

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: