Question 1

To check out whether the regression assumption involving normality of the error terms is valid,it is appropriate to construct a normal probability plot.If this plot forms a straight line from the lower left-hand corner diagonally up to the upper right-hand corner,the error terms may be assumed to be normally distributed.

Accepted Answer

This statement correctly describes how to use a normal probability plot to check the assumption of normality for the error terms in a linear regression model. If the plot shows a straight line, then the error terms are approximately normally distributed.

Question 2

One reason for examining the adjusted R-square value in a multiple regression analysis is that the R-square value will increase just by adding additional independent variables to the model,whereas the adjusted R-square accounts for the relationship between the number of independent variables and the sample size and may actually decline if inappropriate independent variables are included in the model.

Accepted Answer

The statement is correct. The adjusted R-square takes into account the number of independent variables and the sample size, and penalizes for adding irrelevant variables. Thus, it is a better measure of the goodness of fit of the model compared to R-square.

Question 3

It is possible for the standard error of the estimate to actually increase if variables are added to the model that do not aid in explaining the variation in the dependent variable.

Accepted Answer

This can occur if the added variables are not significant or are correlated with existing variables, leading to multicollinearity. When multicollinearity is present, the standard errors increase and the precision of the estimates decreases, which can result in a larger standard error of the estimate.

Question 4

When we say that we wish to determine the aptness of a regression model,we are actually saying that we wish to check to see whether the resulting model meets the basic assumptions of regression analysis.

Accepted Answer

Determining the aptness of a regression model involves checking whether the model meets the basic assumptions of regression analysis.

Question 5

The editors of a national automotive magazine recently studied 30 different automobiles sold in the United States with the intent of seeing whether they could develop a multiple regression model to explain the variation in highway miles per gallon.A number of different independent variables were collected.The following regression output (with some values missing)was recently presented to the editors by the magazine's analysts:  Based on this output and your understanding of multiple regression analysis,how many degrees of freedom are associated with the Residual in the ANOVA table?
A)19
B)22
C)7
D)29

Accepted Answer

The answer of The editors of a national automotive magazine...

Question 6

Which of the following is not an assumption of the multiple regression model?
A)The mean of the residuals is equal to the variance at all combinations of levels of the independent variables.
B)The regression error terms are normally distributed.
C)The model error terms are independent.
D)The residuals have a constant variance for all combinations of values for the independent variables.

Accepted Answer

The assumption that the mean of the residuals is equal to the variance at all combinations of levels of the independent variables is not a valid assumption of the multiple regression model. Instead, the assumption is that the residuals have a mean of zero and a constant variance for all combinations of values for the independent variables.

Question 7

Standard stepwise regression is a good way of identifying potential multicollinearity problems since we are able to see the impact on the model at each step that occurs when a new variable is added to the model.For instance,if bringing in a new variable causes the sign to change on a previously entered variable,we have evidence of multicollinearity.

Accepted Answer

Standard stepwise regression allows us to evaluate the impact of adding new variables to the model and can help identify potential multicollinearity problems. If adding a new variable causes previously entered variable coefficients to change signs or become statistically insignificant, it could indicate multicollinearity.

Question 8

If a stepwise regression approach is used to enter,one at a time,four variables into a regression model,the resulting regression equation may differ from the regression equation that occurs when all four of the variables are entered at one step.

Accepted Answer

The answer of If a stepwise regression approach is used...

Question 9

In a multiple regression model,which of the following is true?
A)The coefficient of determination will be equal to the square of the highest correlation in the correlation matrix.
B)Adding variables that have a low correlation with the dependent variable will cause the R-square value to decline.
C)The sum of the residuals computed for the least squares regression equation will be zero.
D)The adjusted R-square might be higher or lower than the value of the R-square.

Accepted Answer

The answer of In a multiple regression model,which of the...

Question 10

The editors of a national automotive magazine recently studied 30 different automobiles sold in the United States with the intent of seeing whether they could develop a multiple regression model to explain the variation in highway mileage per gallon.A number of different independent variables were collected.The following correlation matrix was developed:  If only one variable were to be brought into the model,which variable should it be if the goal is to explain the highest possible percentage of variation in the dependent variable?
A)0 to 60 mph
B)Horsepower
C)Curb weight
D)Displacement

Accepted Answer

The answer of The editors of a national automotive magazine...

Question 11

The best subsets method will involve trying fewer different regression models than stepwise regression.

Accepted Answer

The best subsets method involves trying all possible combinations of predictors, which can be a much larger number of models compared to stepwise regression, which adds or removes predictors one at a time based on certain criteria.

Question 12

The editors of a national automotive magazine recently studied 30 different automobiles sold in the United States with the intent of seeing whether they could develop a multiple regression model to explain the variation in highway miles per gallon.A number of different independent variables were collected.The following regression output (with some values missing)was recently presented to the editors by the magazine's analysts:  Based on this output and your understanding of multiple regression analysis,what percentage of variation in the dependent variable is explained by the regression model?
A)Approximately 82 percent
B)Over 90 percent
C)About 37 percent
D)None of the above.

Accepted Answer

The coefficient of determination (R^2) is 0.824, which means that approximately 82% of the variation in the dependent variable (highway miles per gallon) is explained by the independent variables included in the multiple regression model.

Question 13

When the best subsets approach is used in a regression application,one method for determining which of the many possible models to select for potential use is called the C_p statistic.

Accepted Answer

The answer of When the best subsets approach is used...

Question 14

The forward selection method and the backward elimination method will always lead to choosing the same final regression model.

Accepted Answer

The forward selection method and the backward elimination method may not always lead to the same final regression model since they use a different approach and may converge on different models.

Question 15

If one or more of the regression assumptions has been violated this means that the current regression model is not the best one for this data set,and another model should be sought.

Accepted Answer

If assumptions are violated, it means that the current model may not be accurate enough and a different model may be needed.

Question 16

The following residual plot is an output of a regression model.  Based on this residual plot,there is evidence to suggest that the underlying relationship between the y variable and the x variable is nonlinear.

Accepted Answer

The residual plot shows a nonlinear pattern, with the residuals increasing and then decreasing as the predicted values increase. This suggests that the relationship between the y variable and the x variable is nonlinear, and not adequately captured by a linear regression model.

Question 17

A useful method for determining whether a linear function is the appropriate function to describe the relationship between the x and y variable is a residual plot in which the residuals are plotted on the vertical axis and the independent variable is on the horizontal axis.

Accepted Answer

This statement is true. Plotting the residuals against the independent variable can help identify any patterns in the data that may indicate a non-linear relationship, and can also help identify any outliers or influential points.

Question 18

The editors of a national automotive magazine recently studied 30 different automobiles sold in the United States with the intent of seeing whether they could develop a multiple regression model to explain the variation in highway miles per gallon.A number of different independent variables were collected.The following regression output (with some values missing)was recently presented to the editors by the magazine's analysts:  Based on this output and your understanding of multiple regression analysis,what is the adjusted R-square value for this model?
A)About 0.82
B)Approximately 0.90
C)Just under 0.48
D)None of the above.

Accepted Answer

The answer of The editors of a national automotive magazine...

Question 19

The following residual plot was constructed based on a simple linear regression model.  Based on this plot,there appears to be no basis for concluding that a curvilinear model may be more appropriate than a linear model to explain the variation in the y variable.

Accepted Answer

The answer of The following residual plot was constructed based...

Question 20

In the best subsets approach to regression analysis,if we start with 4 independent variables,a total of 33 different regression models will actually be computed for possible selection as the best model to use.

Accepted Answer

In the best subsets approach, for $n$ independent variables, $2^n$ different regression models are computed. For 4 independent variables, $2^4 = 16$ different models are computed, not 33.

Quiz 15: Multiple Regression Analysis and Model Building

It is possible for the standard error of the estimate to actually increase if variables are added to the model that do not aid in explaining the variation in the dependent variable.

When we say that we wish to determine the aptness of a regression model,we are actually saying that we wish to check to see whether the resulting model meets the basic assumptions of regression analysis.

Which of the following is not an assumption of the multiple regression model?

If a stepwise regression approach is used to enter,one at a time,four variables into a regression model,the resulting regression equation may differ from the regression equation that occurs when all four of the variables are entered at one step.

In a multiple regression model,which of the following is true?

The best subsets method will involve trying fewer different regression models than stepwise regression.

When the best subsets approach is used in a regression application,one method for determining which of the many possible models to select for potential use is called the Cp statistic.

The forward selection method and the backward elimination method will always lead to choosing the same final regression model.

If one or more of the regression assumptions has been violated this means that the current regression model is not the best one for this data set,and another model should be sought.

The following residual plot is an output of a regression model. Based on this residual plot,there is evidence to suggest that the underlying relationship between the y variable and the x variable is nonlinear.

A useful method for determining whether a linear function is the appropriate function to describe the relationship between the x and y variable is a residual plot in which the residuals are plotted on the vertical axis and the independent variable is on the horizontal axis.

The following residual plot was constructed based on a simple linear regression model. Based on this plot,there appears to be no basis for concluding that a curvilinear model may be more appropriate than a linear model to explain the variation in the y variable.

In the best subsets approach to regression analysis,if we start with 4 independent variables,a total of 33 different regression models will actually be computed for possible selection as the best model to use.

When the best subsets approach is used in a regression application,one method for determining which of the many possible models to select for potential use is called the C_p statistic.