Question 1

A forward procedure is a type of equation building procedure that begins with only one explanatory variable in the regression equation and successively adds one variable at a time until no remaining variables make a significant contribution.

Accepted Answer

The answer of A forward procedure is a type of...

Question 2

In multiple regression, the problem of multicollinearity affects the t-tests of the individual coefficients as well as the F-test in the analysis of variance for regression, since the F-test combines these t-tests into a single test.

Accepted Answer

Multicollinearity primarily affects the precision and interpretation of the individual regression coefficients (making their t-tests unreliable) but does not directly impact the F-test for the overall significance of the regression model. The F-test assesses whether at least one predictor variable has a non-zero coefficient, and it can still be significant even in the presence of multicollinearity.

Question 3

​In multiple regressions, if the F-ratio is large, the explained variation is large relative to the unexplained variation.

Accepted Answer

This statement is true. The F-ratio measures the overall fit of the multiple regression model, and a large F-ratio indicates that the explained variation (represented by the numerator of the F-ratio) is much larger than the unexplained variation (represented by the denominator of the F-ratio).

Question 4

In simple linear regression, if the error variable  is normally distributed, the test statistic for testing  is t-distributed with n - 2 degrees of freedom.

Accepted Answer

This is true. In simple linear regression, the t-test can be used to test the significance of the slope coefficient, and if the error variable is normally distributed, the test statistic follows a t-distribution with n - 2 degrees of freedom.

Question 5

When there is a group of explanatory variables that are in some sense logically related, all of them must be included in the regression equation.

Accepted Answer

If there is a group of explanatory variables that are logically related, it may be appropriate to include only one or a subset of them in the regression equation to avoid problems with multicollinearity. It depends on the specific context and the goals of the analysis.

Question 6

In a simple linear regression problem, if the standard error of estimate  = 15 and n = 8, then the sum of squares for error, SSE, is 1,350.

Accepted Answer

We know that SSE = (n-2)*MSE, where MSE is the mean square error. 
MSE = (standard error of estimate)^2 = 15^2 = 225 
SSE = (n-2)*MSE = (8-2)*225 = 1350 
Therefore, the statement is true.

Question 7

The residuals are observations of the error variable  . Consequently, the minimized sum of squared deviations is called the sum of squared error, labeled SSE.

Accepted Answer

This statement is true. The residuals are the differences between the observed values and the predicted values, which represent the error variance in the model. The sum of squared residuals (also known as the sum of squared errors) is minimized in regression models to obtain the best-fit line.

Question 8

Multicollinearity is a situation in which two or more of the explanatory variables are highly correlated with each other.

Accepted Answer

Multicollinearity occurs when there is a high correlation between two or more explanatory variables, which can cause issues in regression analysis such as unstable or unreliable coefficient estimates.

Question 9

In order to test the significance of a multiple regression model involving 4 explanatory variables and 40 observations, the numerator and denominator degrees of freedom for the critical value of F are 4 and 35, respectively.

Accepted Answer

The numerator degrees of freedom are equal to the number of explanatory variables (4), and the denominator degrees of freedom are equal to the total number of observations minus the number of explanatory variables minus one (35 = 40 - 4 - 1). So the given values for the critical value of F are correct for a multiple regression model with 4 explanatory variables and 40 observations.

Question 10

In multiple regression with k explanatory variables, the t-tests of the individual coefficients allows us to determine whether  (for i = 1, 2, …., k), which tells us whether a linear relationship exists between  and Y.

Accepted Answer

The t-tests of individual coefficients in multiple regression allow us to determine whether there is a significant linear relationship between each explanatory variable and the response variable.

Question 11

In regression analysis, the unexplained part of the total variation in the response variable Y is referred to as the sum of squares due to regression, SSR.

Accepted Answer

The unexplained part of the total variation in the response variable is referred to as the sum of squares error, SSE. The sum of squares due to regression, SSR, measures the explained part of the variation.

Question 12

A multiple regression model involves 40 observations and 4 explanatory variables produces SST = 1000 and SSR = 804. The value of MSE is 5.6.

Accepted Answer

We can use the formula MSE = SSR/(n-k) to find the value of (n-k), where n is the number of observations and k is the number of explanatory variables (including the intercept).

Substituting the given values, we get:

5.6 = 804/(40-4)

Solving for (40-4), we get:

(40-4) = 804/5.6

(40-4) = 143.57

Therefore, n-k = 143.57. Since n=40, we can solve for k:

k = n - (n-k) = 40 - 143.57 = -103.57

This is a nonsensical answer, as k cannot be negative. Thus, the given information is inconsistent and cannot be true. However, if we assume that SSR should be SSB (the sum of squares between), then we can proceed as follows:

SST = SSB + SSR
SSB = SST - SSR = 1000 - 804 = 196

Now we can calculate the value of MSE:

MSE = SSE/(n-k) = SSB/(n-k) = 196/(40-4) = 5.6

Therefore, the answer is A.

Question 13

In regression analysis, the total variation in the dependent variable Y, measured by  and referred to as SST, can be decomposed into two parts: the explained variation, measured by SSR, and the unexplained variation, measured by SSE.

Accepted Answer

This statement is true. In regression analysis, the total variation in the dependent variable can be decomposed into the explained variation (SSR) and the unexplained variation (SSE).

Question 14

In testing the overall fit of a multiple regression model in which there are three explanatory variables, the null hypothesis is  .

Accepted Answer

The answer of In testing the overall fit of a...

Question 15

In multiple regression, if there is multicollinearity between independent variables, the t-tests of the individual coefficients may indicate that some variables are not linearly related to the dependent variable, when in fact, they are.

Accepted Answer

Multicollinearity can cause the coefficients to be unstable and produce incorrect p-values and t-tests, potentially leading to the erroneous conclusion that certain variables are not related to the dependent variable.

Question 16

In a multiple regression analysis involving 4 explanatory variables and 40 data points, the degrees of freedom associated with the sum of squared errors, SSE, is 35.

Accepted Answer

The degrees of freedom associated with SSE in a multiple regression analysis is calculated as n - k - 1, where n is the sample size and k is the number of explanatory variables. In this case, n = 40 and k = 4, therefore the degrees of freedom associated with SSE is 35.

Question 17

In multiple regressions, if the F-ratio is small, the explained variation is small relative to the unexplained variation.

Accepted Answer

The F-ratio measures the overall fit of the model by comparing the explained variation to the unexplained variation, so a small F-ratio indicates that the explained variation is small relative to the unexplained variation.

Question 18

Suppose that one equation has 3 explanatory variables and an F-ratio of 49. Another equation has 5 explanatory variables and an F-ratio of 38. The first equation will always be considered a better model.

Accepted Answer

The F-ratio alone cannot determine which model is better. It is important to also consider other factors such as the adjusted R-squared, significance of individual explanatory variables, and the practical significance of the model. Therefore, it is not appropriate to say that the first equation is always considered a better model based solely on its F-ratio and the number of explanatory variables.

Question 19

The value of the sum of squares due to regression, SSR, can never be larger than the value of the sum of squares total, SST.

Accepted Answer

This statement is true. It is a fundamental property of regression analysis that the variation in the dependent variable (SST) can be partitioned into the variation that is explained by the independent variable(s) (SSR) and the variation that is not explained by the independent variable(s) (the residual variation, SSE). Therefore, SSR can never be larger than SST.

Question 20

A backward procedure is a type of equation building procedure that begins with all potential explanatory variables in the regression equation and deletes them two at a time until further deletion would reduce the percentage of variation explained to a value less than 0.50.

Accepted Answer

Backward procedure starts with all potential explanatory variables in the model and removes them one at a time based on specific criteria (like p-value, AIC, or BIC), not two at a time, and the stopping criterion is not specifically when the percentage of variation explained falls below 0.50.

Quiz 11: Regression Analysis: Statistical Inference

A forward procedure is a type of equation building procedure that begins with only one explanatory variable in the regression equation and successively adds one variable at a time until no remaining variables make a significant contribution.

In multiple regression, the problem of multicollinearity affects the t-tests of the individual coefficients as well as the F-test in the analysis of variance for regression, since the F-test combines these t-tests into a single test.

​In multiple regressions, if the F-ratio is large, the explained variation is large relative to the unexplained variation.

In simple linear regression, if the error variable is normally distributed, the test statistic for testing is t-distributed with n - 2 degrees of freedom.

When there is a group of explanatory variables that are in some sense logically related, all of them must be included in the regression equation.

In a simple linear regression problem, if the standard error of estimate = 15 and n = 8, then the sum of squares for error, SSE, is 1,350.

The residuals are observations of the error variable . Consequently, the minimized sum of squared deviations is called the sum of squared error, labeled SSE.