Question 1

Even when an unimportant variable is added to a regression model,the explained variation will increase.

Accepted Answer

Adding any variable to a regression model will increase the total amount of variation that can be explained by the model, even if the added variable is not significant or relevant to the outcome variable. This increase in variation is represented by the increase in R-squared (or adjusted R-squared) values.

Question 2

When the F test is used to test the overall significance of a multiple regression model,if the null hypothesis is rejected,it can be concluded that all of the independent variables x₁,x₂,…x_k are significantly related to the dependent variable y.

Accepted Answer

Rejecting the null hypothesis in an F test for a multiple regression model only indicates that at least one of the independent variables is significantly related to the dependent variable. It does not conclude that all independent variables are significantly related.

Question 3

The multiple coefficient of determination can assume any value between 0 and 1,inclusive.

Accepted Answer

The range of the multiple coefficient of determination (R-squared) is between 0 and 1, including these values.

Question 4

For the same point estimate of the dependent variable and the same level of significance,the confidence interval is always wider than the corresponding prediction interval.

Accepted Answer

Confidence intervals are always narrower than the corresponding prediction interval.

Question 5

The variance inflation factor measures the relationship between the dependent variable and the rest of the independent variables in the regression model.

Accepted Answer

The variance inflation factor (VIF) measures how much the variance of an estimated regression coefficient increases because of collinearity among the independent variables in the model, not the relationship between the dependent variable and the independent variables.

Question 6

In a regression equation,beta values are previously known parameter values.

Accepted Answer

Beta values are actually the estimated coefficients of the independent variables in a regression equation, calculated based on the data. They are not previously known parameter values.

Question 7

For combinations of data within the experimental region,the least squares plane is the estimate of the plane of means.

Accepted Answer

The least squares plane is a statistical method used to estimate the plane that fits the data best in the experimental region. It is calculated using the mean of the response variable for each combination of the independent variables. Therefore, it is the estimate of the plane of means.

Question 8

When the quadratic regression model y = $\beta$ 0+ $\beta$ 1x + $\beta$ 2x²+ ? is used,the term $\beta$ 1 shows the rate of curvature of the parabola.

Accepted Answer

The term $\beta_2$ in the quadratic regression model $y = \beta_0 + \beta_1x + \beta_2x^2$ shows the rate of curvature of the parabola, not $\beta_1$. $\beta_1$ represents the linear component of the model.

Question 9

Testing the contribution of individual independent variables with t-tests is performed prior to the F-test for the model in multiple regression analysis.

Accepted Answer

In multiple regression analysis, the F-test for the overall model typically precedes individual t-tests for the contribution of each independent variable, as the F-test assesses whether at least one predictor variable has a statistically significant relationship with the dependent variable, setting the stage for further investigation into specific relationships.

Question 10

An outlier may be an incorrectly entered data value.

Accepted Answer

Outliers are data values that are significantly different from others in the dataset. Sometimes, outliers can occur due to errors in data entry or measurement, making them incorrect values that should be removed or corrected.

Question 11

If we increase the number of independent variables in a multiple regression model,the F statistic will always increase.

Accepted Answer

The F statistic in a multiple regression model does not always increase with the addition of more independent variables. The F statistic is designed to test the overall significance of the model, and while adding variables can sometimes increase the F statistic if those variables significantly improve the model's fit, it can also decrease or remain unchanged if the added variables do not contribute to explaining the variability in the dependent variable.

Question 12

In a regression model,at any given combination of values of the independent variables,the population of potential error terms is assumed to have an F-distribution.

Accepted Answer

In a regression model, at any given combination of values of the independent variables, the population of potential error terms is assumed to have a normal distribution, not an F-distribution. The F-distribution is used in hypothesis testing for comparing variances or comparing the fit of different regression models.

Question 13

A t-test is used in testing the significance of an individual independent variable.

Accepted Answer

For testing the significance of any independent variable, use a t-test.

Question 14

Due to the fact that multiple regression models consist of multiple independent variables,residual analysis cannot be performed.

Accepted Answer

Residual analysis can be performed in multiple regression models to assess the degree of variability in the response variable that is not explained by the independent variables.

Question 15

The error term in the regression model describes the effects of all factors other than the independent variables on y (response variable).

Accepted Answer

The error term captures the variation in y that cannot be explained by the independent variables. These unobserved factors or sources of randomness are collectively referred to as the error term, and it may include measurement error, omitted variables, and other types of stochastic variability.

Question 16

The standard error decreases if and only if the adjusted multiple coefficient of determination decreases.

Accepted Answer

There is no direct relationship between the standard error and the adjusted multiple coefficient of determination. The standard error may decrease due to different reasons like an increase in sample size or better predictive variables whereas the adjusted multiple coefficient of determination depends on the number of predictors in the model and their correlation with the outcome variable.

Question 17

Regression models that employ more than one independent variable are referred to as multiple regression models.

Accepted Answer

This statement is true. Multiple regression models use two or more independent variables to predict the dependent variable.

Question 18

An application of the multiple regression model generated the following results involving the F test of the overall regression model: p - value = .0012,R² = .67 and s = .076.Thus,the null hypothesis,which states that none of the independent variables are significantly related to the dependent variable,should be rejected at the .01 level of significance.

Accepted Answer

The p-value is less than the level of significance (.0012 < .01), thus we reject the null hypothesis and conclude that at least one independent variable is significantly related to the dependent variable.

Question 19

In a regression model,a value of the error term depends upon other values of the error term.

Accepted Answer

In a properly specified regression model, the error terms are assumed to be independent of each other, meaning the value of one error term does not depend on the values of other error terms.

Question 20

The normal plot is a residual plot that checks the assumption that,for any point in the experimental region,the population of potential error terms is normally distributed.

Accepted Answer

The normal plot, also known as a Q-Q plot or quantile-quantile plot, is a graphical tool used to check for normality in the distribution of residuals. It compares the distribution of the residuals to a theoretical normal distribution. If the residuals are normally distributed, the plot will show a straight line pattern. Therefore, the normal plot helps to verify the assumption that for any point in the experimental region, the population of potential error terms is normally distributed.

Quiz 12: Multiple Regression and Model Building