Deck 13: Multiple Regression

Question 1

If R² and R²_adj differ greatly, we should probably add a few predictors to improve the fit.

Answer 1

False

Question 2

A common misinterpretation of the principle of Occam's Razor is that a simple regression model (rather than a multiple regression model) is always best.

Answer 2

True

Question 3

A parsimonious model is one with many weak predictors but a few strong ones.

Answer 3

False

Question 4

For a certain firm, the regression equation Bonus = 2,000 + 257 Experience + 0.046 Salary describes employee bonuses with a standard error of 125. John has 10 years' experience, earns $50,000, and earned a bonus of $7,000. John is an outlier.

Question 5

Evans' Rule says that if n = 50 you need at least 5 predictors to have a good model.

Question 6

A predictor whose pairwise correlation with Y is near zero can still have a significant t-value in a multiple regression when other predictors are included.

Question 7

The model Y = β₀ + β₁X + β₂X² cannot be estimated by Excel because of the nonlinear term.

Question 8

If a regression model's F test statistic is F_calc = 43.82, we could say that the explained variance is approximately 44 percent.

Question 9

In a multiple regression with 3 predictors in a sample of 25 U.S. cities, we would use F_3,21 in a test of overall significance.

Question 10

A negative estimated coefficient in a regression usually indicates a weak predictor.

Question 11

The random error term in a regression model reflects all factors omitted from the model.

Question 12

The effect of a binary predictor is to shift the regression intercept.

Question 13

If the probability plot of residuals resembles a straight line, the residuals show a fairly good fit to the normal distribution.

Question 14

R²_adj can exceed R² if there are several weak predictors.

Question 15

A binary (categorical) predictor should not be used along with nonbinary (numerical) predictors.

Question 16

The F statistic in a multiple regression is significant if at least one of the predictors has a significant t statistic at a given α.

Question 17

There is one residual for each predictor in the regression model.

Question 18

Confidence intervals for Y may be unreliable when the residuals are not normally distributed.

Question 19

In a regression, the model with the best fit is preferred over all other models.

Question 20

In regression the dependent variable is referred to as the response variable.

Question 21

Plotting the residuals against a binary predictor (X = 0, 1) reveals nothing about heteroscedasticity.

Question 22

If the residuals in your regression are non-normal, a larger sample size might help improve the reliability of confidence intervals for Y.

Question 23

A disadvantage of Excel's Data Analysis regression tool is that it expects the independent variables to be in a block of contiguous columns, so you must delete a column if you want to eliminate a predictor from the model.

Question 24

Multicollinearity can be detected from t tests of the predictor variables.

Question 25

The regression equation Bonus = 2,812 + 27 Experience + 0.046 Salary says that Experience is the most significant predictor of Bonus.

Question 26

When autocorrelation is present, the estimates of the coefficients will be unbiased.

Question 27

Non-normality of the residuals from a regression can best be detected by looking at the residual plots against the fitted Y values.

Question 28

The F statistic and its p-value give a global test of significance for a multiple regression.

Question 29

A disadvantage of Excel's regression is that it does not give as much accuracy in the estimated regression coefficients as a package like Minitab.

Question 30

In a regression model of student grades, we would code the nine categories of business courses taken (ACC, FIN, ECN, MGT, MKT, MIS, ORG, POM, QMM) by including nine binary (0 or 1) predictors in the regression.

Question 31

Statisticians who work with cross-sectional data generally do not anticipate autocorrelation.

Question 32

A high variance inflation factor (VIF) indicates a significant predictor in the regression.

Question 33

A multiple regression with 60 observations should not have 13 predictors.

Question 34

A regression of Y using four independent variables X₁, X₂, X₃, X₄ could also have up to four nonlinear terms (X²) and six simple interaction terms (X_jX_k) if you have enough observations to justify them.

Question 35

The first differences transformation might be tried if autocorrelation is found in a time-series data set.

Question 36

The ill effects of heteroscedasticity might be mitigated by redefining totals (e.g., total number of homicides) as relative values (e.g., homicide rate per 100,000 population).

Question 37

When multicollinearity is present, the regression model is of no use for making predictions.

Question 38

Autocorrelation may be detected by looking at a plot of the residuals against time.

Question 39

Autocorrelation of the residuals may affect the reliability of the t-values for the estimated coefficients of the predictors X₁, X_2,…, X_k.

Question 40

A widening pattern of residuals as X increases would suggest heteroscedasticity.

Question 41

A test is conducted in 22 cities to see if giving away free transit system maps will increase the number of bus riders. In a regression analysis, the dependent variable Y is the increase in bus riders (in thousands of persons) from the start of the test until its conclusion. The independent variables are X₁ = the number (in thousands) of free maps distributed and a binary variable X₂ = 1 if the city has free downtown parking, 0 otherwise. The estimated regression equation is $\hat { Y } = 1.32 + 0.0345 X _ { 1 } - 1.45 X _ { 2 }$ . In city 3, the observed Y value is 7.3, X₁ = 140, and X₂ = 0. The residual for city 3 (in thousands) is:

A)6.15.
B)1.15.
C)4.83.
D)1.57.

Question 42

In a multiple regression with five predictors in a sample of 56 U.S. cities, we would use F_5,50 in a test of overall significance.

Question 43

Non-normal residuals lead to biased estimates of the coefficients in a regression model.

Question 44

Unlike other predictors, a binary predictor has a t-value that is either 0 or 1.

Question 45

Multicollinearity refers to relationships among the independent variables.

Question 46

A fitted multiple regression equation is Y = 12 + 3X₁ - 5X₂ + 7X₃ + 2X₄. When X₁ increases 2 units and X₂ increases 2 units as well, while X₃ and X₄ remain unchanged, what change would you expect in your estimate of Y?

A)Decrease by 2
B)Decrease by 4
C)Increase by 2
D)No change in Y

Question 47

A multiple regression analysis with two independent variables yielded the following results in the ANOVA table: SS(Total) = 798, SS(Regression) = 738, SS(Error) = 60. The multiple correlation coefficient is:

A).2742
B).0752
C).9248
D).9617

Question 48

A large VIF (e.g., 10 or more) would indicate multicollinearity.

Question 49

If X₂ is a binary predictor in Y = β₀ + β₁X₁ + β₂X₂, then which statement is most nearly correct?

A)X₂ = 1 should represent the most desirable condition.
B)X₂ would be a significant predictor if β₂ = 423.72.
C)X₂ = 0, X₂ = 1, X₂ = 2 would be appropriate if three categories exist.
D)X₂ will shift the estimated equation either by 0 units or by β₂ units.

Question 50

The unexplained sum of squares measures variation in the dependent variable Y about the:

A)mean of the Y values.
B)estimated Y values.
C)mean of the X values.
D)Y-intercept.

Question 51

In a multiple regression with five predictors in a sample of 56 U.S. cities, what would be the critical value for an F-test of overall significance at α = .05?

A)2.45
B)2.37
C)2.40
D)2.56

Question 52

The t test shows the ratio of an estimated coefficient to its standard error.

Question 53

In a multiple regression with six predictors in a sample of 67 U.S. cities, what would be the critical value for an F-test of overall significance at α = .05?

A)2.29
B)2.25
C)2.37
D)2.18

Question 54

Heteroscedasticity exists when all the errors (residuals) have the same variance.

Question 55

Which of the following is not true of the standard error of the regression?

A)It is a measure of the accuracy of the prediction.
B)It is based on squared vertical deviations between the actual and predicted values of Y.
C)It would be negative when there is an inverse relationship in the model.
D)It is used in constructing confidence and prediction intervals for Y.

Question 56

Non-normality of residuals is not usually considered a major problem unless there are outliers.

Question 57

When predictor variables are strongly related to each other, the __________ of the regression estimates is questionable.

A)logic
B)fit
C)parsimony
D)stability

Question 58

In the fitted regression Y = 12 + 3X₁ - 5X₂ + 27X₃ + 2X₄ the most significant predictor is X₃.

Question 59

Given that the fitted regression is Y = 76.40 - 6.388X₁ + 0.870X₂, the standard error of b₁ is 1.453, and n = 63, at α = .05, we can conclude that X₁ is a significant predictor of Y.

Question 60

A squared predictor is used to test for nonlinearity in the predictor's relationship to Y.

Question 61

Part of a regression output is provided below. Some of the information has been omitted. $$\begin{array} { l c c c c } \hline \text { Source of variation } & S S & d f & M S & F \\\hline \text { Regression } & 3177.17 & 2 & 1588.6 & \\\text { Residual } & & 17 & 17.717 & \\\text { Total } & 3478.36 & 19 & & \\\hline\end{array}$$ The approximate value of F is:

A)1605.7.
B)0.9134.
C)89.66.
D)impossible to calculate with the given information.

Question 62

Which is a characteristic of the variance inflation factor (VIF)?

A)It is insignificant unless the corresponding t statistic is significant.
B)It reveals collinearity rather than multicollinearity.
C)It measures the degree of significance of each predictor.
D)It indicates the predictor's degree of multicollinearity.

Question 63

Part of a regression output is provided below. Some of the information has been omitted. $$\begin{array} { l c c c c } \hline \text { Source of variation } & S S & d f & M S & F \\\hline \text { Regression } & 3177.17 & 2 & 1588.6 & \\\text { Residual } & & 17 & 17.717 & \\\text { Total } & 3478.36 & 19 & & \\\hline\end{array}$$ The SS (residual) is:

A)3177.17.
B)301.19.
C)17.71.
D)impossible to determine.

Question 64

A fitted multiple regression equation is Y = 28 + 5X₁ - 4X₂ + 7X₃ + 2X₄. When X₁ increases 2 units and X₂ increases 2 units as well, while X₃ and X₄ remain unchanged, what change would you expect in your estimate of Y?

A)Increase by 2
B)Decrease by 4
C)Increase by 4
D)No change in Y

Question 65

Using a sample of 63 observations, a dependent variable Y is regressed against two variables X₁ and X₂ to obtain the fitted regression equation Y = 76.40 - 6.388X₁ + 0.870X₂. The standard error of b₁ is 3.453 and the standard error of b₂ is 0.611. At α = .05, we could:

A)conclude that both coefficients differ significantly from zero.
B)reject H₀: β₁ ≥ 0 and conclude H₀: β₁ < 0.
C)reject H₀: β₂ ≤ 0 and conclude H₀: β₁ > 0.
D)conclude that Evans' Rule has been violated.

Question 66

Refer to this ANOVA table from a regression: $$\begin{array} { l r r r r } \hline \text { Source } & d f & \text { SS } & M S & F \\\hline \text { Regression } & 4 & 1793.2356 & 448.3089 & 7.48540 \\\text { Residual } & 45 & 2695.0996 & 59.8911 & \\\text { Total } & 49 & 4488.3352 & & \\\hline\end{array}$$ Which statement is not accurate?

A)The F-test is significant at ? = .05.
B)There were 50 observations.
C)There were 5 predictors.
D)There would be 50 residuals.

Question 67

A realtor is trying to predict the selling price of houses in Greenville (in thousands of dollars) as a function of Size (measured in thousands of square feet) and whether or not there is a fireplace (FP is 0 if there is no fireplace, 1 if there is a fireplace). Part of the regression output is provided below, based on a sample of 20 homes. Some of the information has been omitted. $$\begin{array} { l r r r r } \hline \text { Variable } & \text { Coefficients } & \text { Standard Error } & \text { t-Stat } & \text { P-value } \\\hline \text { Intercept } & 128.93746 & 2.6205302 & 49.203 & 8.93 \mathrm { E } - 20 \\\text { Size } & & 1.2072436 & 11.439 & 2.09 \mathrm { E } - 09 \\\text { FP } & 6.47601954 & 1.9803612 & 3.27 & 0.004512 \\\hline\end{array}$$ Which statement is supported by the regression output?

A)At ? = .05, FP is not a significant predictor in a two-tailed test.
B)A fireplace adds around $6476 to the selling price of the average house.
C)A large house with no fireplace will sell for more than a small house with a fireplace.
D)FP is a more significant predictor than Size.

Question 68

A useful guideline in determining the extent of collinearity in a multiple regression model is:

A)Sturges' Rule.
B)Klein's Rule.
C)Occam's Rule.
D)Pearson's Rule.

Question 69

Refer to the following correlation matrix that was part of a regression analysis. The dependent variable was Abort (the number of abortions per 1000 women of childbearing age). The regression was estimated using data for the 50 U.S. states with these predictors: EdSpend = public K-12 school expenditure per capita, Age = median age of population, Unmar = percent of total births by unmarried women, Infmor = infant mortality rate in deaths per 1000 live births. $$\begin{array}{l}\text { Correlation Matrix }\\\begin{array} { l c c c c c } \hline & \text { Abort } & \text { EdSpend } & \text { Age } & \text { Unmar } & \text { InfMor } \\\hline \text { Abort } & 1.0000 & & & & \\\text { EdSpend } & 0.2626 & 1.0000 & & & \\\text { Age } & 0.1610 & - 0.0420 & 1.0000 & & \\\text { Unmar } & 0.3286 & - 0.0949 & 0.0937 & 1.0000 & \\\text { InfMor } & - 0.2513 & - 0.2826 & 0.0389 & 0.5239 & 1.0000 \\\hline\end{array}\end{array}$$ Using a two-tailed correlation test, which statement is not accurate?

A)Age and Infmor are not significantly correlated at ? = .05.
B)Abort and Unmar are significantly correlated at ? = .05.
C)Unmar and Infmor are significantly correlated at ? = .05.
D)The first column of the table shows evidence of multicollinearity.

Question 70

In a multiple regression all of the following are true regarding residuals except:

A)their sum always equals zero.
B)they are the differences between observed and predicted values of the response variable.
C)they may be used to detect multicollinearity.
D)they may be used to detect heteroscedasticity.

Question 71

The residual plot below suggests which violation(s) of regression assumptions?

A)Autocorrelation
B)Heteroscedasticity
C)Non-normality
D)Multicollinearity

Question 72

Refer to this ANOVA table from a regression: $$\begin{array} { l r r r r } \hline \text { Source } & d f & \text { SS } & M S & F \\\hline \text { Regression } & 4 & 1793.2356 & 448.3089 & 7.48540 \\\text { Residual } & 45 & 2695.0996 & 59.8911 & \\\text { Total } & 49 & 4488.3352 & & \\\hline\end{array}$$ For this regression, the R² is:

A).3995.
B).6005.
C).6654.
D).8822.

Question 73

A realtor is trying to predict the selling price of houses in Greenville (in thousands of dollars) as a function of Size (measured in thousands of square feet) and whether or not there is a fireplace (FP is 0 if there is no fireplace, 1 if there is a fireplace). Part of the regression output is provided below, based on a sample of 20 homes. Some of the information has been omitted. $$\begin{array} { l r r r } \hline \text { Variable } & \text { Coefficients } & \text { Standard Error } & t \text {-Stat } \\\hline \text { Intercept } & 128.93746 & 2.6205302 & 49.203 \\\text { Size } & & 1.2072436 & 11.439 \\\text { FP } & 6.47601954 & 1.9803612 & 3.27 \\\hline\end{array}$$ The estimated coefficient for Size is approximately:

A)9.5.
B)13.8.
C)122.5.
D)1442.6.

Question 74

Which is not a standard criterion for assessing a regression model?

A)Logic of causation
B)Overall fit
C)Degree of collinearity
D)Binary predictors

Question 75

A log transformation might be appropriate to alleviate which problem(s)?

A)Heteroscedastic residuals
B)Multicollinearity
C)Autocorrelated residuals

Question 76

A realtor is trying to predict the selling price of houses in Greenville (in thousands of dollars) as a function of Size (measured in thousands of square feet) and whether or not there is a fireplace (FP is 0 if there is no fireplace, 1 if there is a fireplace). The regression output is provided below. Some of the information has been omitted. $$\begin{array} { l c c c c } \hline \text { Source of variation } & \text { SS } & d f & M S & F \\\hline \text { Regression } & 3177.17 & 2 & 1588.6 & \\\text { Residual } & & 17 & 17.717 & \\\text { Total } & 3478.36 & 19 & & \\\hline\end{array}$$ How many predictors (independent variables) were used in the regression?

A)20
B)18
C)3
D)2

Question 77

Refer to the following regression results. The dependent variable is Abort (the number of abortions per 1000 women of childbearing age). The regression was estimated using data for the 50 U.S. states with these predictors: EdSpend = public K-12 school expenditure per capita, Age = median age of population, Unmar = percent of total births by unmarried women, Infmor = infant mortality rate in deaths per 1000 live births. $$\begin{array} { l r r r } \hline \text { Variable } & \text { Coefficients } & \text { Standard Error } & t - \text { Stat } \\\hline \text { Intercept } & - 19.244 & 25.387 & - 0.758 \\\text { EdSpend } & 0.0080040 & 0.0054472 & 1.469 \\\text { Age } & 0.76012 & 0.66914 & 1.136 \\\text { Unmar } & 0.98629 & 0.22132 & 4.456 \\\text { InfMor } & - 3.7848 & 1.0173 & - 3.720 \\\hline\end{array}$$ Which statement is not supported by a two-tailed test?

A)Unmar is a significant predictor at ? = .01.
B)EdSpend is a significant predictor at ? = .20.
C)Infmor is not a significant predictor at ? = .05.
D)Age is not a significant predictor at ? = .05.

Question 78

A realtor is trying to predict the selling price of houses in Greenville (in thousands of dollars) as a function of Size (measured in thousands of square feet) and whether or not there is a fireplace (FP is 0 if there is no fireplace, 1 if there is a fireplace). The regression output is provided below. Some of the information has been omitted. $$\begin{array} { l c r c c } \hline \text { Source of variation } & \text { SS } & d f & \text { MS } & F \\\hline \text { Regression } & 3177.17 & 2 & 1588.584 & \\\text { Residual } & & 17 & 17.71713 & \\\text { Total } & 3478.36 & 19 & & \\\hline\end{array}$$ Which of the following conclusions can be made based on the F-test?

A)The p-value on the F-test will be very high.
B)At least one of the predictors is useful in explaining Y.
C)The model is of no use in predicting selling prices of houses.
D)The estimates were based on a sample of 19 houses.

Question 79

If the standard error is 12, the width of a quick prediction interval for Y is:

A)±15.
B)±24.
C)±19.
D)impossible to determine without an F table.

Question 80

Which is not a name often given to an independent variable that takes on just two values (0 or 1) according to whether or not a given characteristic is absent or present?

A)Absent variable
B)Binary variable
C)Dummy variable