Question 1

TABLE 15-1
A certain type of rare gem serves as a status symbol for many of its owners.In theory,for low prices,the demand increases and it decreases as the price of the gem increases.However,experts hypothesize that when the gem is valued at very high prices,the demand increases with price due to the status owners believe they gain in obtaining the gem.Thus,the model proposed to best explain the demand for the gem by its price is the quadratic model:
Y = β₀ + β₁X + β₁X² + ε
where Y = demand (in thousands)and X = retail price per carat.
This model was fit to data collected for a sample of 12 rare gems of this type.A portion of the computer analysis obtained from Microsoft Excel is shown below:
 
-Referring to Table 15-1,does there appear to be significant upward curvature in the response curve relating the demand (Y)and the price (X)at 10% level of significance?
A)Yes,since the p-value for the test is less than 0.10.
B)No,since the value of β is near 0.
C)No,since the p-value for the test is greater than 0.10.
D)Yes,since the value of β is positive.

Accepted Answer

C

Question 2

The Variance Inflationary Factor (VIF)measures the correlation of the X variables with the Y variable.

Accepted Answer

The Variance Inflationary Factor (VIF) measures the multicollinearity between the X variables in a multiple regression model, not their correlation with the Y variable.

Question 3

TABLE 15-1
A certain type of rare gem serves as a status symbol for many of its owners.In theory,for low prices,the demand increases and it decreases as the price of the gem increases.However,experts hypothesize that when the gem is valued at very high prices,the demand increases with price due to the status owners believe they gain in obtaining the gem.Thus,the model proposed to best explain the demand for the gem by its price is the quadratic model:
Y = β₀ + β₁X + β₁X² + ε
where Y = demand (in thousands)and X = retail price per carat.
This model was fit to data collected for a sample of 12 rare gems of this type.A portion of the computer analysis obtained from Microsoft Excel is shown below:
 
-Referring to Table 15-1,a more parsimonious simple linear model is likely to be statistically superior to the fitted curvilinear for predicting sale price (Y).

Accepted Answer

True

Question 4

TABLE 15-1
A certain type of rare gem serves as a status symbol for many of its owners.In theory,for low prices,the demand increases and it decreases as the price of the gem increases.However,experts hypothesize that when the gem is valued at very high prices,the demand increases with price due to the status owners believe they gain in obtaining the gem.Thus,the model proposed to best explain the demand for the gem by its price is the quadratic model:
Y = β₀ + β₁X + β₁X² + ε
where Y = demand (in thousands)and X = retail price per carat.
This model was fit to data collected for a sample of 12 rare gems of this type.A portion of the computer analysis obtained from Microsoft Excel is shown below:
 
-Referring to Table 15-1,what is the correct interpretation of the coefficient of multiple determination?
A)98.8% of the total variation in demand can be explained by the linear relationship between demand and price.
B)98.8% of the total variation in demand can be explained by the quadratic relationship between demand and price.
C)98.8% of the total variation in demand can be explained by the addition of the square term in price.
D)98.8% of the total variation in demand can be explained by just the square term in price.

Accepted Answer

The coefficient of multiple determination is the R-squared value, which represents the proportion of total variation in the dependent variable (demand) that can be explained by the independent variable(s) (in this case, price and the square of price). A multiple R-squared value of 0.988 or 98.8% indicates that almost all of the variation in demand can be attributed to the quadratic relationship between demand and price. Therefore, option B is the correct interpretation of the coefficient of multiple determination.

Question 5

A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies.She proceeds to randomly select 26 large corporations and record information in millions of dollars.A statistical analyst discovers that capital spending by corporations has a significant inverse relationship with wage spending.What should the microeconomist who developed this multiple regression model be particularly concerned with?
A)Randomness of error terms
B)Collinearity
C)Normality of residuals
D)Missing observations

Accepted Answer

The significant inverse relationship between capital spending and wage spending suggests that there may be collinearity issues in the regression model, where both variables are highly correlated with each other. This can lead to unreliable coefficient estimates and difficulty in interpreting the results. The microeconomist should investigate this issue further and consider possible solutions, such as dropping one of the variables or transforming the data. While randomness of error terms, normality of residuals, and missing observations are important considerations in regression analysis, they are not the primary concern in this scenario.

Question 6

In multiple regression,the ________ procedure permits variables to enter and leave the model at different stages of its development.
A)forward selection
B)residual analysis
C)backward elimination
D)stepwise regression

Accepted Answer

Stepwise regression allows variables to enter and leave the model based on a predetermined criterion, such as significance level or coefficient size. This approach can help to identify the most important variables in a model and avoid overfitting. Forward selection only allows variables to enter the model, while backward elimination only allows variables to leave the model. Residual analysis is a diagnostic tool used to check the assumptions of regression, but it does not determine variable selection.

Question 7

A real estate builder wishes to determine how house size (House)is influenced by family income (Income),family size (Size),and education of the head of household (School).House size is measured in hundreds of square feet,income is measured in thousands of dollars,and education is in years.The builder randomly selected 50 families and constructed the multiple regression model.The business literature involving human capital shows that education influences an individual's annual income.Combined,these may influence family size.With this in mind,what should the real estate builder be particularly concerned with when analyzing the multiple regression model?
A)Randomness of error terms
B)Collinearity
C)Normality of residuals
D)Missing observations

Accepted Answer

The real estate builder should be particularly concerned with collinearity between the independent variables. This is because education is also a predictor of family income, which may overlap with the influence of family income on house size. Collinearity can lead to unstable and unreliable regression coefficients, making it difficult to accurately interpret the relationships between the independent and dependent variables.

Question 8

The Variance Inflationary Factor (VIF)measures the
A)correlation of the X variables with the Y variable.
B)correlation of the X variables with each other.
C)contribution of each X variable with the Y variable after all other X variables are included in the model.
D)standard deviation of the slope.

Accepted Answer

VIF measures the correlation between predictor variables (X variables) in a regression model. High VIF values indicate a high degree of correlation between two or more predictor variables, which can lead to multicollinearity and affect the reliability of the regression model. Therefore, to avoid multicollinearity, it is recommended to remove the predictor variables with high VIF values. The VIF value is calculated for each predictor variable using the formula 1 / (1 - R^2), where R^2 is the coefficient of determination obtained by regressing that predictor variable against all other predictor variables in the model.

Question 9

TABLE 15-1
A certain type of rare gem serves as a status symbol for many of its owners.In theory,for low prices,the demand increases and it decreases as the price of the gem increases.However,experts hypothesize that when the gem is valued at very high prices,the demand increases with price due to the status owners believe they gain in obtaining the gem.Thus,the model proposed to best explain the demand for the gem by its price is the quadratic model:
Y = β₀ + β₁X + β₁X² + ε
where Y = demand (in thousands)and X = retail price per carat.
This model was fit to data collected for a sample of 12 rare gems of this type.A portion of the computer analysis obtained from Microsoft Excel is shown below:
 
-Referring to Table 15-1,what is the p-value associated with the test statistic for testing whether there is an upward curvature in the response curve relating the demand (Y)and the price (X)?
A)0.0001
B)0.0006
C)0.3647
D)None of the above.

Accepted Answer

The answer of TABLE 15-1
A certain type of rare gem...

Question 10

TABLE 15-2
In Hawaii,condemnation proceedings are under way to enable private citizens to own the property that their homes are built on.Until recently,only estates were permitted to own land,and homeowners leased the land from the estate.In order to comply with the new law,a large Hawaiian estate wants to use regression analysis to estimate the fair market value of the land.The following model was fit to data collected for n = 20 properties,10 of which are located near a cove.
 
-Referring to Table 15-2,is the overall model statistically adequate at a 0.05 level of significance for predicting sale price (Y)?
A)No,since some of the t tests for the individual variables are not significant.
B)No,since the standard deviation of the model is fairly large.
C)Yes,since none of the β-estimates are equal to 0.
D)Yes,since the p-value for the test is smaller than 0.05.

Accepted Answer

The question asks whether the overall model is statistically adequate for predicting sale price at a 0.05 level of significance. The best way to answer this is by looking at the p-value for the test, which is provided in the ANOVA table in Table 15-2. The p-value for the model is 0.002, which is smaller than 0.05, indicating that the model is statistically significant at this level of significance. Therefore, the correct answer is D.

Question 11

The Cp statistic is used
A)to determine if there is a problem of collinearity.
B)if the variances of the error terms are all the same in a regression model.
C)to choose the best model.
D)to determine if there is an irregular component in a time series.

Accepted Answer

The Cp statistic is used to choose the best model in regression analysis. It evaluates the overall fit of the model by considering both the goodness of fit and the number of variables in the model. Therefore, it is useful in selecting the most appropriate model among a set of candidate models. The other options (A, B, and D) are not correct because they refer to different statistical concepts or methods.

Question 12

TABLE 15-2
In Hawaii,condemnation proceedings are under way to enable private citizens to own the property that their homes are built on.Until recently,only estates were permitted to own land,and homeowners leased the land from the estate.In order to comply with the new law,a large Hawaiian estate wants to use regression analysis to estimate the fair market value of the land.The following model was fit to data collected for n = 20 properties,10 of which are located near a cove.
 
-Referring to Table 15-2,given a quadratic relationship between sale price (Y)and property size (X₁),what null hypothesis would you test to determine whether the curves differ from cove and non-cove properties?
A)H₀: β₂ = β₃ = β₅ = 0
B)H₀: β₄ = β₅ = 0
C)H₀: β₃ = β₅ = 0
D)H₀: β₂ = 0

Accepted Answer

The null hypothesis that would test whether the curves for sale price vs. property size differ between cove and non-cove properties would be to set the coefficients of the interaction terms between cove and the quadratic term and the linear term to zero. In this case, that would be β₃ = β₅ = 0. Therefore, the correct choice is C.

Question 13

TABLE 15-1
A certain type of rare gem serves as a status symbol for many of its owners.In theory,for low prices,the demand increases and it decreases as the price of the gem increases.However,experts hypothesize that when the gem is valued at very high prices,the demand increases with price due to the status owners believe they gain in obtaining the gem.Thus,the model proposed to best explain the demand for the gem by its price is the quadratic model:
Y = β₀ + β₁X + β₁X² + ε
where Y = demand (in thousands)and X = retail price per carat.
This model was fit to data collected for a sample of 12 rare gems of this type.A portion of the computer analysis obtained from Microsoft Excel is shown below:
 
-Referring to Table 15-1,what is the value of the test statistic for testing whether there is an upward curvature in the response curve relating the demand (Y)and the price (X)?
A)-5.14
B)0.95
C)373
D)None of the above.

Accepted Answer

The answer of TABLE 15-1
A certain type of rare gem...

Question 14

If independent variables are not significant individually but are significant as a group at a specified level of significance,this is most likely due to
A)autocorrelation.
B)the presence of dummy variables.
C)the absence of dummy variables.
D)collinearity.

Accepted Answer

This is most likely due to collinearity, which occurs when two or more independent variables are highly correlated with each other, making it difficult to determine the individual impact of each variable on the dependent variable. As a result, they may not be significant individually but become significant as a group.

Question 15

Which of the following is used to find a "best" model?
A)Odds ratio
B)Mallow's Cp
C)Standard error of the estimate
D)SST

Accepted Answer

Mallow's Cp is used to find the "best" model by determining the smallest number of predictor variables that still provide a good fit to the data. A lower Cp value indicates a better model. The other options listed do not directly measure the fit or goodness of a model.

Question 16

As a project for his business statistics class,a student examined the factors that determined parking meter rates throughout the campus area.Data were collected for the price per hour of parking,blocks to the quadrangle,and one of the three jurisdictions: on campus,in downtown and off campus,or outside of downtown and off campus.The population regression model hypothesized is Yᵢ = β₀ + β₁X₁ᵢ + β₂X₂ᵢ + β₃X₃ᵢ + ε where
Y is the meter price
X₁ is the number of blocks to the quad
X₂ is a dummy variable that takes the value 1 if the meter is located in downtown and off campus and the value 0 otherwise
X₃ is a dummy variable that takes the value 1 if the meter is located outside of downtown and off campus,and the value 0 otherwise
Suppose that whether the meter is located on campus is an important explanatory factor.Why should the variable that depicts this attribute not be included in the model?
A)Its inclusion will introduce autocorrelation.
B)Its inclusion will introduce collinearity.
C)Its inclusion will inflate the standard errors of the estimated coefficients.
D)Both B and C.

Accepted Answer

Including the variable for whether the meter is located on campus would introduce collinearity with the other dummy variables for downtown and off campus and outside of downtown and off campus. This would inflate the standard errors of the estimated coefficients and make it difficult to interpret the impact of each variable on the meter price. Therefore, it is best to leave out this variable in the regression model.

Question 17

Collinearity is present when there is a high degree of correlation between the dependent variable and any of the independent variables.

Accepted Answer

Collinearity refers to a situation where two or more independent variables in a regression model are highly correlated, not between the dependent variable and an independent variable.

Question 18

A regression diagnostic tool used to study the possible effects of collinearity is
A)the slope.
B)the Y-intercept.
C)the VIF.
D)the standard error of the estimate.

Accepted Answer

Collinearity occurs when predictor variables in a model are highly correlated with each other. VIF (Variance Inflation Factor) is a regression diagnostic tool used to study the possible effects of collinearity. It measures the magnitude of multicollinearity and ranges from 1 (no multicollinearity) to infinity (perfect multicollinearity). A VIF value greater than 10 is a concern for multicollinearity. Therefore, C is the best choice.

Question 19

Collinearity is present when there is a high degree of correlation between independent variables.

Accepted Answer

Collinearity occurs when there is a strong linear relationship between two or more independent variables in a regression model. This can result in unstable estimates of the regression coefficients and decreased model interpretability.

Question 20

TABLE 15-2
In Hawaii,condemnation proceedings are under way to enable private citizens to own the property that their homes are built on.Until recently,only estates were permitted to own land,and homeowners leased the land from the estate.In order to comply with the new law,a large Hawaiian estate wants to use regression analysis to estimate the fair market value of the land.The following model was fit to data collected for n = 20 properties,10 of which are located near a cove.
 
-Referring to Table 15-2,given a quadratic relationship between sale price (Y)and property size (X₁),what test should be used to test whether the curves differ from cove and non-cove properties?
A)F test for the entire regression model
B)t test on each of the coefficients in the entire regression model
C)Partial F test on the subset of the appropriate coefficients
D)t test on each of the subsets of the appropriate coefficients

Accepted Answer

The question is asking to test if the curves differ between cove and non-cove properties, which implies a comparison of the coefficients associated with the variable indicating location near the cove. Therefore, a partial F test on the subset of the appropriate coefficients would be the best choice. This test would allow us to see if the coefficient for the variable indicating proximity to the cove is significant and if its inclusion in the model improves the fit compared to a model without that variable. A t test on each subset of the appropriate coefficients would not have the same power as a partial F test, and testing each coefficient separately with a t test does not take into account the joint significance of the variables in the model. Testing the entire regression model with an F test or testing each coefficient separately with a t test would not specifically address the question of interest.

Quiz 15: Multiple Regression Model Building

The Variance Inflationary Factor (VIF)measures the correlation of the X variables with the Y variable.

In multiple regression,the ________ procedure permits variables to enter and leave the model at different stages of its development.

The Variance Inflationary Factor (VIF) measures the

The Cp statistic is used

If independent variables are not significant individually but are significant as a group at a specified level of significance,this is most likely due to

Which of the following is used to find a "best" model?

Collinearity is present when there is a high degree of correlation between the dependent variable and any of the independent variables.

A regression diagnostic tool used to study the possible effects of collinearity is

Collinearity is present when there is a high degree of correlation between independent variables.