Question 1

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

Accepted Answer

Given the model, a 5 unit increase in $x_2$ (while holding $x_1$ and $x_3$ constant) would result in a decrease in $\hat{y}$ because the coefficient for $x_2$ is -15. Thus, a 5 unit increase in $x_2$ would lead to a decrease of $5 \times -15 = -75$ in the predicted value of $y$.

Question 2

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 coefficient of determination will be:

Accepted Answer

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

Question 3

In a regression model involving 60 observations, the following estimated regression model was obtained: $\hat { y }$ = 51.4 + 0.70x1 + 0.679 x2 - 0.378 x3. For this model, total variation in y = SST = 119,724 and SSR = 29,029.72. The value of MSE is:

Accepted Answer

MSE is calculated as (SST - SSR) / (n - k - 1), where n is the number of observations and k is the number of predictors. Here, MSE = (119,724 - 29,029.72) / (60 - 3 - 1) = 1619.541.

Question 4

In testing the validity of a multiple regression model involving 5 independent variables and 30 observations, the numbers of degrees of freedom for the numerator and denominator (respectively) for the critical value of F will be:

Accepted Answer

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

Question 5

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

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

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:

Accepted Answer

The coefficient of determination, R^2, represents the proportion of the total variation in the dependent variable that is accounted for by the regression equation. In this case, R^2 = 0.64, which means that 64% of the variation in y is explained by the regression equation. Therefore, the correct answer is D.

Question 7

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:

Accepted Answer

MSE (mean squared error) is calculated as SSE (sum of squared errors) divided by the degrees of freedom (n-k-1), where n is the number of observations and k is the number of predictors. In this case, n=50 and k=3, so degrees of freedom = 50-3-1 = 46.
MSE = SSE/46 = 175/46 = 3.804. Therefore, the answer is D.

Question 8

Which of the following best describes first-order autocorrelation?

Accepted Answer

First-order autocorrelation is a condition in which a relationship exists between consecutive residuals. This means that the value of one residual is related to the value of the preceding residual. It is also known as lag 1 autocorrelation. It can be a problem in statistical analyses as it violates the assumption of independence of residuals.

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

The null hypothesis in the context of testing the validity of a multiple regression model posits that the coefficients of the independent variables ($\beta_1, \beta_2, \beta_3, \beta_4$) have no effect on the dependent variable, which is mathematically represented as all these coefficients being equal to zero.

Question 11

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:

Accepted Answer

For a sample size of 30, the degrees of freedom for testing a regression coefficient is $n - 2 = 30 - 2 = 28$. At a 5% significance level, the critical value for a two-tailed test from a t-distribution table with 28 degrees of freedom is approximately 2.056.

Question 12

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:

Accepted Answer

The degrees of freedom for testing individual coefficients in multiple regression analysis with k independent variables and n observations is equal to n-k-1. Thus, in this case, the degrees of freedom would be 110-9-1=100. Therefore, the critical value of t for testing individual coefficients in the model will have 100 degrees of freedom.

Question 13

In a regression model involving 30 observations, the following estimated regression model was obtained: $\hat { y }$ = 60 + 2.8x1 + 1.2 x2 - x3. For this model, total variation in y = SST = 800 and SSE = 200. The value of the F-statistic for testing the validity of this model is:

Accepted Answer

The F-statistic is calculated as $ F = \frac{MSR}{MSE} $, where $ MSR = \frac{SSR}{k} $ and $ MSE = \frac{SSE}{n-k-1} $. Here, $ SSR = SST - SSE = 800 - 200 = 600 $, $ k = 3 $, and $ n = 30 $. Thus, $ MSR = \frac{600}{3} = 200 $ and $ MSE = \frac{200}{26} \approx 7.69 $. Therefore, $ F = \frac{200}{7.69} \approx 26.00 $.

Question 14

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

Accepted Answer

Multicollinearity occurs when the independent variables are highly correlated with each other, not with the dependent variable.

Question 15

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

Accepted Answer

In multiple regression, there are multiple independent variables used to predict the dependent variable.

Question 16

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

Accepted Answer

The Durbin-Watson test is used to determine if there is first-order autocorrelation, which is when there is a correlation between a variable and its lagged values in a time series analysis. It is not used to test for multicollinearity, heteroscedasticity, or homoscedasticity.

Question 17

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:

Accepted Answer

The formula for calculating the multiple standard error of estimate (MSE) is:

MSE = sqrt(SSres/(n-k-1))

Where SSres is the sum of squared residuals (in this case, it is given as 180), n is the number of data points (20), and k is the number of independent variables (4).

Plugging in these values, we get:

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

Therefore, the multiple standard error of estimate is approximately 3.464. Option B is the correct answer.

Question 18

In a multiple regression analysis involving 20 observations and 5 independent variables, total variation in y = SST = 250 and SSE = 35. The coefficient of determination adjusted for degrees of freedom is:

Accepted Answer

The answer of In a multiple regression analysis involving 20...

Question 19

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 is used when there are more than one independent variables that are believed to influence the dependent variable. It estimates the linear relationship between the dependent variable and all the independent variables together.

Question 20

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

Accepted Answer

The formula for calculating the F-value in a multiple regression analysis is:

F = [(SSR/p)/(SSE/(n-p-1))]

where p is the number of independent variables and n is the sample size.

In this case, SSR = 300, p = 5, SSE = 200, and n = 25.

F = [(300/5)/(200/(25-5-1))] = 5.70

Therefore, the F-value of the test statistic will be 5.70.

Quiz 16: Multiple Regression

For the multiple regression model y^\hat { y }y^​ = 75 + 25x1 - 15 x2 + 10 x3, if x2x _ { 2 }x2​ were to increase by 5, holding x1x _ { 1 }x1​ and x3x _ { 3 }x3​ constant, the value of y would: