Question 1

How do you go about checking for multicollinearity?

Accepted Answer

The answer of How do you go about checking for...

Question 2

A high correlation between two independent variables is an indication of ____________________.

Accepted Answer

The answer of A high correlation between two independent variables...

Question 3

One clue to the presence of multicollinearity is an independent variable known to be an important predictor that ends up having a regression coefficient that is not ____________________.

Accepted Answer

The answer of One clue to the presence of multicollinearity...

Question 4

Three predictor variables are being considered for use in a linear regression model. Given the correlation matrix below, does it appear that multicollinearity could be a problem?
 $$\begin{array} { | l | c c c | } 
\hline  & \boldsymbol { x } _ { 1 } & \boldsymbol { x } _ { \mathbf { 1 } } & \boldsymbol { x } _ { \mathbf { 3 } } \
\hline \boldsymbol { x } _ { 1 } & 1.000 & & \
\boldsymbol { x } _ { \mathbf { 2 } } & 0.025 & 1.000 & \
\boldsymbol { x } _ { \mathbf { 3 } } & 0.968 & 0.897 & 1.000 \
\hline
\end{array}$$

Accepted Answer

The answer of Three predictor variables are being considered for...

Question 5

In reference to the equation $\hat { y } = 1.86 - 0.51 x _ { 1 } + 0.60 x _ { 2 }$ , the value 0.60 is the average change in y per unit change in x₂, regardless of the value of x₁.

Accepted Answer

The answer of In reference to the equation \(\hat {...

Question 6

In multiple regression analysis, the adjusted coefficient of determination is adjusted for the number of independent variables and the sample size.

Accepted Answer

The adjusted coefficient of determination adjusts for the number of independent variables in addition to the sample size in multiple regression analysis.

Question 7

In a multiple regression analysis involving 50 observations and 5 independent variables, the total variation in y is 475 and SSE = 71.25. Then, the coefficient of determination is 0.85.

Accepted Answer

The coefficient of determination (R-squared) is the proportion of the total variation in the dependent variable (y) that is explained by the independent variables in the model. It is calculated as:

R-squared = 1 - (SSE/SST)

where SSE is the sum of squared errors (also known as residual sum of squares), and SST is the total sum of squares, which is the sum of squared deviations of y from its mean. Therefore, we have:

SST = total variation in y = 475
SSE = 71.25

Substituting these values into the formula, we get:

R-squared = 1 - (71.25/475) = 0.85

Therefore, the coefficient of determination is indeed 0.85.

Question 8

A practical way to identify multicollinearity is through the examination of a correlation ____________________ that shows the correlations of each variable with each of the other variables.

Accepted Answer

The answer of A practical way to identify multicollinearity is...

Question 9

A multiple regression model involves 40 observations and 4 independent variables produces a total variation in y of 100,000 and SSR = 80,400. Then, the value of MSE is 560.

Accepted Answer

MSE = SSR/(n-p-1) = 80,400/(40-4-1) = 2,800

This is different from the given value of MSE in the question, which is 560. Therefore, the statement is false.

Question 10

In testing the significance of a multiple regression model with three independent variables, the null hypothesis is $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 }$ .

Accepted Answer

The null hypothesis for testing the significance of a multiple regression model is $H_0: \beta_1 = \beta_2 = \beta_3 = 0$, indicating that none of the independent variables have an effect.

Question 11

In reference to the equation $\hat { y } = - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value -0.80 is the y-intercept.

Accepted Answer

The value -0.80 is the constant term in the equation, representing the y-intercept when all other variables ($x_1$ and $x_2$) are zero.

Question 12

In reference to the equation $\hat { y } = - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value 0.12 is the average change in y per unit change in x₁, when x₂ is held constant.

Accepted Answer

The coefficient 0.12 represents the partial derivative of $\hat{y}$ with respect to $x_1$, indicating the average change in $\hat{y}$ for a one-unit increase in $x_1$ while holding $x_2$ constant.

Question 13

There are several clues to the presence of multicollinearity. One clue is when a regression coefficient exhibits the wrong ____________________.

Accepted Answer

The answer of There are several clues to the presence...

Question 14

A multiple regression equation has a coefficient of determination of 0.81. Then, the percentage of the variation in y that is explained by the regression equation is 90%.

Accepted Answer

The percentage of variation in y that is explained by the regression equation is equal to the coefficient of determination (R-squared) times 100. Therefore, if the R-squared is 0.81, the percentage of variation explained is 81%, not 90%.

Question 15

In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the numerator and denominator degrees of freedom for the critical value of F are 3 and 21, respectively.

Accepted Answer

The numerator degrees of freedom for the F-test in a multiple regression model are based on the number of independent variables, which in this case is 4. Therefore, the numerator degrees of freedom should be 4. The denominator degrees of freedom are calculated as the total number of observations minus the number of independent variables minus 1 (N - k - 1), which would be 25 - 4 - 1 = 20, not 21.

Question 16

There are several clues to the presence of multicollinearity. One clue is when an independent variable is added or deleted, the regression coefficients for the other variables ____________________.

Accepted Answer

The answer of There are several clues to the presence...

Question 17

Discuss two indicators that can be found in an analysis that suggest multicollinearity is present.

Accepted Answer

The answer of Discuss two indicators that can be found...

Question 18

An adverse effect of multicollinearity is that the estimated regression coefficients of the independent variables that are correlated tend to have large sampling ____________________.

Accepted Answer

The answer of An adverse effect of multicollinearity is that...

Question 19

In a multiple regression analysis involving 4 independent variables and 30 data points, the number of degrees of freedom associated with the sum of squares for error, SSE, is 25.

Accepted Answer

The number of degrees of freedom associated with SSE in a multiple regression analysis is calculated as n - p - 1, where n is the number of data points and p is the number of independent variables. Therefore, in this case, 30 - 4 - 1 = 25 degrees of freedom are associated with SSE.

Question 20

Most statistical software print a second R² statistic, called the coefficient of determination adjusted for degrees of freedom, which has been adjusted to take into account the sample size and the number of independent variables.

Accepted Answer

The statement is true. The adjusted coefficient of determination is a commonly reported statistic in statistical software output and is calculated by adjusting the coefficient of determination (R-squared) to account for the number of independent variables and the sample size.

Quiz 17: Multiple Regression