Multiple Regression Model Building 538 CHAPTER 15: MULTIPLE REGRESSION MODEL BUILDING 1. 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 ran the multiple regression. 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 ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: collinearity, assumption 2. 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 ANSWER: b
Transcript
Multiple Regression Model Building 538
CHAPTER 15: MULTIPLE REGRESSION MODEL BUILDING
1. 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 ran the multiple regression. 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 termsb) Collinearityc) Normality of residualsd) Missing observations
ANSWER:bTYPE: MC DIFFICULTY: ModerateKEYWORDS: collinearity, assumption
2. 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 termsb) Collinearityc) Normality of residualsd) Missing observations
ANSWER:bTYPE: MC DIFFICULTY: ModerateKEYWORDS: collinearity, assumption
3. In multiple regression, the __________ procedure permits variables to enter and leave the model at different stages of its development.
a) forward selectionb) residual analysisc) backward eliminationd) stepwise regression
ANSWER:dTYPE: MC DIFFICULTY: Easy KEYWORDS: stepwise regression, model building
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4. A regression diagnostic tool used to study the possible effects of collinearity isa) the slope.b) the Y-intercept.c) the VIF.d) the standard error of the estimate.
ANSWER:cTYPE: MC DIFFICULTY: Easy KEYWORDS: variance inflationary factor, collinearity
5. Which of the following is not used to find a "best" model?a) Adjusted r2
b) Mallow's Cp
c) Odds ratiod) All of the above
ANSWER:cTYPE: MC DIFFICULTY: Moderate KEYWORDS: model building
6. 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.
ANSWER:bTYPE: MC DIFFICULTY: Easy KEYWORDS: variance inflationary factor, collinearity
7. The 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.
ANSWER:cTYPE: MC DIFFICULTY: EasyKEYWORDS: C-p statistic, model building
TABLE 15-1
To explain personal consumption (CONS) measured in dollars, data is collected for
INC: personal income in dollarsCRDTLIM: $1 plus the credit limit in dollars available to the individual
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APR: average annualized percentage interest rate for borrowing for the individualADVT: per person advertising expenditure in dollars by manufacturers in the city
where the individual livesSEX: gender of the individual; 1 if female, 0 if male
A regression analysis was performed with CONS as the dependent variable and ln(CRDTLIM), ln(APR), ln(ADVT), and SEX as the independent variables. The estimated model was
8. Referring to Table 15-1, what is the correct interpretation for the estimated coefficient for ADVT?
a) A $1 increase in per person advertising expenditure by the manufacturer will result in an estimated average increase of $2.35 on personal consumption holding other variables constant.
b) A 100% increase in per person advertising expenditure by the manufacturer will result in an estimated average increase of $2.35 on personal consumption holding other variables constant.
c) A 100% increase in per person advertising expenditure by the manufacturer will result in an estimated average increase of 2.35% on personal consumption holding other variables constant.
d) A 1% increase in per person advertising expenditure by the manufacturer will result in an estimated average increase of 2.35% on personal consumption holding other variables constant.
ANSWER:bTYPE: MC DIFFICULTY: DifficultKEYWORDS: transformation, slope, interpretation
9. Referring to Table 15-1, what is the correct interpretation for the estimated coefficient for APR?a) A one percentage point increase in average annualized percentage interest rate will result
in an estimated average increase of $5.77 on personal consumption holding other variables constant.
b) A 100% increase in average annualized percentage interest rate will result in an estimated average increase of 5.77% on personal consumption holding other variables constant.
c) A 100% increase in average annualized percentage interest rate will result in an estimated average increase of $5.77 on personal consumption holding other variables constant.
d) A 1% increase in average annualized percentage interest rate will result in an estimated average increase of 5.77% on personal consumption holding other variables constant.
ANSWER:cTYPE: MC DIFFICULTY: ModerateKEYWORDS: transformation, slope, interpretation
TABLE 15-2
A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand 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
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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 0 1 X 2 X 2 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:
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.994R Square 0.988Standard Error 12.42Observations 12
Coeff StdError t Stat P-valueIntercept 286.42 9.66 29.64 0.0001Price – 0.31 0.06 – 5.14 0.0006Price Sq 0.000067 0.00007 0.95 0.3647
10. Referring to Table 15-2, 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) -5.14b) 0.95c) 373d) None of the above.
ANSWER:bTYPE: MC DIFFICULTY: EasyKEYWORDS: quadratic regression, t test on slope, test statistic
11. Referring to Table 15-2, 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) 0.0001b) 0.0006c) 0.3647d) None of the above.
ANSWER:cTYPE: MC DIFFICULTY: EasyKEYWORDS: quadratic regression, t test on slope, p-value
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12. Referring to Table 15-2, 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.
ANSWER:bTYPE: MC DIFFICULTY: ModerateKEYWORDS: coefficient of multiple determination, interpretation
13. Referring to Table 15-2, 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 2 is near 0.c) No, since the p-value for the test is greater than 0.10.d) Yes, since the value of 2 is positive.
ANSWER:cTYPE: MC DIFFICULTY: ModerateKEYWORDS: quadratic regression, t test on slope, decision, conclusion
14. True or False: Referring to Table 15-2, a more parsimonious simple linear model is likely to be statistically superior to the fitted curvilinear for predicting sale price (Y).
ANSWER:TrueTYPE: TF DIFFICULTY: ModerateKEYWORDS: quadratic regression, t test on slope, interpretation
TABLE 15-3
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.
Model 1: Y 0 1 X1 2 X2 3 X1 X2 4 X12 5 X1
2 X2 where Y = Sale price of property in thousands of dollars
X1 = Size of property in thousands of square feetX2 = 1 if property located near cove, 0 if not
Using the data collected for the 20 properties, the following partial output obtained from Microsoft Excel is shown:
SUMMARY OUTPUT
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Regression StatisticsMultiple R 0.985R Square 0.970Standard Error 9.5Observations 20
15. Referring to Table 15-3, 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 -estimates are equal to 0.d) Yes, since the p-value for the test is smaller than 0.05.
ANSWER:dTYPE: MC DIFFICULTY: EasyKEYWORDS: F test on the entire regression, p-value, decision, conclusion
16. Referring to Table 15-3, 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?
ANSWER:cTYPE: MC DIFFICULTY: DifficultKEYWORDS: interaction, partial F test, form of hypothesis
17. Referring to Table 15-3, 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.
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d) t test on each of the subsets of the appropriate coefficients.
ANSWER:cTYPE: MC DIFFICULTY: DifficultKEYWORDS: interaction, partial F test, interpretation
18. If a group of 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.
ANSWER:dTYPE: MC DIFFICULTY: EasyKEYWORDS: collinearity, assumption, properties
19. 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 where Y is the meter pricex1 is the number of blocks to the quadx2 is a dummy variable that takes the value 1 if the meter is located in downtown and off campus and the value 0 otherwisex3 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).
ANSWER:dTYPE: MC DIFFICULTY: ModerateKEYWORDS: dummy variable, collinearity, properties
20. True or False: The Variance Inflationary Factor (VIF) measures the correlation of the X variables with the Y variable.
22. True or False: Collinearity is present when there is a high degree of correlation between the dependent variable and any of the independent variables.
23. True or False: A high value of R2 significantly above 0 in multiple regression accompanied by insignificant t-values on all parameter estimates very often indicates a high correlation between independent variables in the model.
24. True or False: One of the consequences of collinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.
29. True or False: Two simple regression models were used to predict a single dependent variable. Both models were highly significant, but when the two independent variables were placed in the same multiple regression model for the dependent variable, R2 did not increase substantially and the parameter estimates for the model were not significantly different from 0. This is probably an example of collinearity.
30. True or False: So that we can fit curves as well as lines by regression, we often use mathematical manipulations for converting one variable into a different form. These manipulations are called dummy variables.
A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a “centered” curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been “centered.”
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.747R Square 0.558Adjusted R Square 0.478Standard Error 863.1
Coeff StdError t Stat P-valueIntercept 1283.0 352.0 3.65 0.0040CenDose 25.228 8.631 2.92 0.0140CenDoseSq 0.8604 0.3722 2.31 0.0410
31. Referring to Table 15-4, the prediction of time to relief for a person receiving a dose of the drug 10 units above the average dose (i.e., the prediction of Y for X = 10), is ________.
ANSWER:1,621 TYPE: FI DIFFICULTY: Moderate KEYWORDS: quadratic regression, prediction of individual values
32. Referring to Table 15-4, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. The p-value of the test is ________.
ANSWER:0.041TYPE: FI DIFFICULTY: Difficult KEYWORDS: quadratic regression, partial F test, p-value
33. Referring to Table 15-4, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. The value of the test statistic is ________.
ANSWER:2.312 or 5.3361TYPE: FI DIFFICULTY: Moderate KEYWORDS: quadratic regression, partial F test, test statistic
34. True or False: Referring to Table 15-4, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. If she chooses to use a level of significance of 0.05, she would decide that there is a significant curvilinear relationship.
35. True or False: Referring to Table 15-4, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. If she chooses to use a level of significance of 0.01 she would decide that there is a significant curvilinear relationship.
36. Referring to Table 15-4, suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a curvilinear model that includes a linear term. The p-value of the test statistic for the contribution of the curvilinear term is ________.
ANSWER:0.041TYPE: FI DIFFICULTY: Moderate KEYWORDS: quadratic regression, t test on slope, p-value
37. Referring to Table 15-4, suppose the chemist decides to use a t test to determine if there is a significant difference between a curvilinear model without a linear term and a curvilinear model that includes a linear term. The value of the test statistic is ______.
ANSWER:2.92TYPE: FI DIFFICULTY: Moderate KEYWORDS: quadratic regression, t test on slope, test statistic
38. Referring to Table 15-4, suppose the chemist decides to use a t test to determine if there is a significant difference between a curvilinear model without a linear term and a curvilinear model that includes a linear term. The p-value of the test is ______.
ANSWER:0.0140TYPE: FI DIFFICULTY: Moderate KEYWORDS: quadratic regression, t test on slope, p-value
39. True or False: Referring to Table 15-4, suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a curvilinear model that includes a linear term. If she used a level of significance of 0.05, she would decide that the linear model is sufficient.
ANSWER:FalseTYPE: TF DIFFICULTY: Moderate KEYWORDS: quadratic regression, t test on slope, decision
40. True or False: Referring to Table 15-4, suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a curvilinear model that includes a linear term. If she used a level of significance of 0.02, she would decide that the linear model is sufficient.
ANSWER:TrueTYPE: TF DIFFICULTY: Moderate KEYWORDS: quadratic regression, t test on slope, decision
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41. True or False: Referring to Table 15-4, suppose the chemist decides to use a t test to determine if there is a significant difference between a curvilinear model without a linear term and a curvilinear model that includes a linear term. Using a level of significance of 0.05, she would decide that the curvilinear model should include a linear term.
ANSWER:TrueTYPE: TF DIFFICULTY: Moderate KEYWORDS: quadratic regression, t test on slope, decision
42. In multiple regression, the __________ procedure permits variables to enter and leave the model at different stages of its development.
ANSWER:stepwise regressionTYPE: FI DIFFICULTY: Easy KEYWORDS: stepwise regression
43. A regression diagnostic tool used to study the possible effects of collinearity is ______.
ANSWER:VIFTYPE: FI DIFFICULTY: Moderate KEYWORDS: collinearity, variance inflationary factor
44. The _______ (larger/smaller) the value of the Variance Inflationary Factor, the higher is the collinearity of the X variables.
ANSWER:largerTYPE: FI DIFFICULTY: Moderate KEYWORDS: variance inflationary factor, collinearity, properties
45. The logarithm transformation can be useda) to overcome violations to the autocorrelation assumption.b) to test for possible violations to the autocorrelation assumption.c) to overcome violations to the homoscedasticity assumption.d) to test for possible violations to the homoscedasticity assumption.
ANSWER:cTYPE: MC DIFFICULTY: ModerateKEYWORDS: transformation, homoscedasticity, assumption
46. The logarithm transformation can be useda) to overcome violations to the autocorrelation assumption.b) to test for possible violations to the autocorrelation assumption.c) to change a nonlinear model into a linear model.d) to change a linear independent variable into a nonlinear independent variable.
ANSWER:c
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TYPE: MC DIFFICULTY: ModerateKEYWORDS: transformation, autocorrelation, homoscedasticity, assumption
47. Which of the following will NOT change a nonlinear model into a linear model?a) Quadratic regression modelb) Logarithmic transformationc) Square-root transformationd) Variance inflationary factor
ANSWERS:dTYPE: MC DIFFICULTY: EasyKEYWORDS: transformation
48. An independent variable Xj is considered highly correlated with the other independent variables ifa)
b)
c) for
d) for
ANSWER:bTYPE: MC DIFFICULTY: EasyKEYWORDS: variance inflationary factor, collinearity
49. True or False: The goals of model building are to find a good model with the fewest independent variables that is easier to interpret and has lower probability of collinearity.
ANSWER:TrueTYPE: TF DIFFICULTY: EasyKEYWORDS: model building, collinearity
50. Using the best-subsets approach to model building, models are being considered when theira)
b)
c)
d)
ANSWER:dTYPE: MC DIFFICULTY: EasyKEYWORDS: model building, C-p statistic
51. True or False: In data mining where huge data sets are being explored to discover relationships among a large number of variables, the best-subsets approach is more practical than the stepwise regression approach.
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ANSWER:FalseTYPE: TF DIFFICULTY: EasyKEYWORDS: model building, stepwise regression, Cp
52. True or False: The stepwise regression approach takes into consideration all possible models.
ANSWER:FalseTYPE: TF DIFFICULTY: EasyKEYWORDS: model building, stepwise regression
53. True or False: In stepwise regression, an independent variable is not allowed to be removed from the model once it has entered into the model.
ANSWER:FalseTYPE: TF DIFFICULTY: EasyKEYWORDS: model building, stepwise regression
54. True or False: Using the Cp statistic in model building, all models with are equally good.
ANSWER:FalseTYPE: TF DIFFICULTY: EasyKEYWORDS: model building, Cp
TABLE 15-8
The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state.Let Y = % Passing as the dependent variable, =:% Attendance, = Salaries and = Spending.
The coefficient of multiple determination ( ) of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743.
The output from the best-subset regressions is given below:Adjusted
63. Referring to Table 15-5, what is the value of the test statistic to determine whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance?
ANSWER:2.2792TYPE: PR DIFFICULTY: EasyKEYWORDS: quadratic regression, model building, t test on slope, test statistic
64. Referring to Table 15-5, what is the p-value of the test statistic to determine whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance?
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ANSWER:0.0276TYPE: PR DIFFICULTY: EasyKEYWORDS: quadratic regression, model building, t test on slope, p-value
65. True or False: Referring to Table 15-5, the null hypothesis should be rejected when testing whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance.
ANSWER:TrueTYPE: TF DIFFICULTY: EasyKEYWORDS: quadratic regression, model building, t test on slope, decision
66. True or False: Referring to Table 15-5, the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is not significant at a 5% level of significance.
ANSWER:FalseTYPE: TF DIFFICULTY: EasyKEYWORDS: quadratic regression, model building, t test on slope, conclusion
