Deck 12: Multiple Regression

Question 1

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A real estate broker is interested in identifying the factors that determine the price of a house.She wants to run the following regression: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε where Y = price of the house in $1,000s,X₁ = number of bedrooms,X₂ = square footage of living space,and X₃ = number of miles from the beach.Taking a sample of 30 houses,the broker runs a multiple regression and gets the following results:
= 123.2 + 4.59x₁ + 0.125x₂ - 6.04x₃,
= 103.2,
= 2.13,
= 0.062,
= 4.17,R² = 0.47,and
= 0.45 (adjusted).
What should the null and alternative hypotheses be for β₂?

A)H₀ : β₂ = 0, H₁ : β₂ > 0
B)H₀ : β₂ ≠ 0, H₁ : β₂ = 0
C)H₀ : β₂ > 0, H₁ : β₂ < 0
D)H₀ : β₂ = 0, H₁ : β₂ ≠ 0

Answer 1

H₀ : β₂ = 0, H₁ : β₂ > 0

Question 2

Determine the price that an individual has to pay for a 3 bedroom,1,000 square foot house that is located three miles away from the beach.

A)$201,422
B)$177,243
C)$243,850
D)$229,198

Answer 2

$243,850

Question 3

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
In examining the determinants of income,data were collected regarding the characteristics of 45 adults,and the regression Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ +ε was used,where Y is the annual income (in thousands of dollars),X₁ is the person's age,X₂ is his/her years of education,and X₃ is a dummy variable = 1 if the adult is female.
If you get = 26.3 + 1.38x₁ + 2.98x₂ - 0.76x₃ + 0.34(x₂ ∙ x₃)when you run the regression,how would you interpret the coefficient on his/her years of education?

A)For each additional year of education,women on average earn $420 less than men.
B)For each additional year of education,women approximately earn $1,100 less than men.
C)For each additional year of education,women on average earn an additional $3,320.
D)For each additional year of education,women approximately earn $340 less than men.

Answer 3

For each additional year of education,women on average earn $420 less than men.

Question 4

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A real estate broker is interested in identifying the factors that determine the price of a house.She wants to run the following regression: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε where Y = price of the house in $1,000s,X₁ = number of bedrooms,X₂ = square footage of living space,and X₃ = number of miles from the beach.Taking a sample of 30 houses,the broker runs a multiple regression and gets the following results:
= 123.2 + 4.59x₁ + 0.125x₂ - 6.04x₃,
= 103.2,
= 2.13,
= 0.062,
= 4.17,R² = 0.47,and
= 0.45 (adjusted).
What should the null and alternative hypotheses be for β₁?

A)H₀ : β₁ ≠ 0, H₁ : β₁ = 0
B)H₀ : β₁ = 0, H₁ : β₁ < 0
C)H₀ : β₁ = 0, H₁ : β₁ > 0
D)H₀ : β₁ < 0, H₁ : β₁ > 0

Question 5

What is your conclusion regarding the hypothesis test for β₃?

A)fail to reject H₀ at α ≥ 0.10
B)reject H₀ at α = 0.05
C)reject H₀ at α = 0.025
D)reject H₀ at α = 0.10

Question 6

You are interested in determining the factors that determine the wealth of a person and assume that the age of a person may have a non-linear effect on wealth.You run the following regression: Y = β₀ + β₁X₁ + β₂X₂ + β₃


+ β₄X₃ + ε where Y is the total wealth and X₂ is the person's age.What would be your H₀ regarding β₂ and β₃?

A)β₂ > 0 and β₃ > 0
B)β₂ < 0 and β₃ < 0
C)β₂ > 0 and β₃ < 0
D)β₂ < 0 and β₃ > 0

Question 7

Which of the following would you use to apply multiple regression to situations when independent variables are categorical?

A)correlated variables
B)transformed variables
C)dependent variables
D)dummy variables

Question 8

What would you conclude if you fail to reject H₀ : β₁ = β₂ = ∙ ∙ ∙ ∙ ∙ ∙ = β_k = 0?

A)No relationship exists between the dependent variable and the independent variables.
B)A strong relationship exists among the independent variables.
C)The independent variables are good predictors of the dependent variable.
D)More information is needed to answer the question.

Question 9

What is the correct interpretation of the coefficient of determination R²?

A)The sales price equals 53% of these three variables.
B)47% of the variation in sales prices can be determined by the variation in these three variables.
C)The sales price equals 47% of these three variables.
D)The probability that the sample values will lie on the regression line is 0.47.

Question 10

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
In examining the determinants of income,data were collected regarding the characteristics of 45 adults,and the regression Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ +ε was used,where Y is the annual income (in thousands of dollars),X₁ is the person's age,X₂ is his/her years of education,and X₃ is a dummy variable = 1 if the adult is female.
If you get = 26.3 + 1.38x₁ + 2.98x₂ - 0.76x₃ when you run the regression,how would you interpret the coefficient on gender?

A)A woman earns 76% of a man's earnings.
B)For each additional year of education,a woman earns $760 less than a man.
C)For each additional year in the age,a woman earns $760 less than a man.
D)On average,a woman earns $760 less than a man.

Question 11

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A real estate broker is interested in identifying the factors that determine the price of a house.She wants to run the following regression: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε where Y = price of the house in $1,000s,X₁ = number of bedrooms,X₂ = square footage of living space,and X₃ = number of miles from the beach.Taking a sample of 30 houses,the broker runs a multiple regression and gets the following results:
= 123.2 + 4.59x₁ + 0.125x₂ - 6.04x₃,
= 103.2,
= 2.13,
= 0.062,
= 4.17,R² = 0.47,and
= 0.45 (adjusted).
What should the null and alternative hypotheses be for β₃?

A)H₀ : β₃ = 0,H₁ : β₃ > 0
B)H₀ : β₃ = 0,H₁ : β₃ < 0
C)H₀ : β₃ ≠ 0, H₁ : β₃ = 0
D)H₀ : β₃ < 0,H₁ : β₃ > 0

Question 12

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
In examining the determinants of income,data were collected regarding the characteristics of 45 adults,and the regression lnY = β₀ + β₁ lnX₁ + β₂ lnX₂ + β₃X₃ + ε was used,where Y is the annual income (in thousands of dollars),X₁ is the adult's age,X₂ is his/her years of education,and X₃ is a dummy
variable = 1 and is used if the adult is female.You run the regression and obtain the equation
ln
= 6.3 + 0.91 lnx₁ + 1.3 ln x₂ - 0.05x₃.
How would you interpret the coefficient on years of education?

A)For every addition in his/her years of education,we would expect the income to increase on average by $1,310,assuming that all the other independent variables in the model are held constant.
B)For every one percent increase in his/her years of education,we would expect the income to increase on average by 1.3 percent,assuming that all the other independent variables in the model are held constant.
C)For every 1.3 percent increase in his/her years of education,we would expect the income to increase approximately by $1,000,assuming that all the other independent variables in the model are held constant.
D)For every one percent increase in his/her years of education,we would expect the income to increase approximately by $1,310,assuming that all the other independent variables in the model are held constant.

Question 13

How would you interpret the coefficient on x₃?

A)We would expect to pay approximately $1,000 more for a house that is located 6 miles away from the beach.
B)We would expect to pay approximately $6,040 more for a house that is located an additional mile away from the beach.
C)We would expect to pay,on average,$1,000 less for a house that is located 6 miles away from the beach,given that the number of bedrooms and square footage of living space are held constant.
D)We would expect to pay,on average,$6,040 less for a house that is located an additional mile away from the beach,given that the number of bedrooms and square footage of living space are held constant.

Question 14

The regressions Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε and Y = β₀ + β₁X₁ + β₂X₂ + ε were run using a sample of 30 observations.The SSE for the first regression is 298.4 and 382.3 for the second regression.Test H₀ : β₃ = β₄ = 0.

A)Reject H₀ at α = 0.05
B)Reject H₀ at α = 0.025
C)Reject H₀ at α = 0.01
D)Fail to reject H₀ at α < 0.10

Question 15

The standard error of the estimate for a multiple regression model with two explanatory variables X₁ and X₂:

A)measures the variation around the predicted regression equation.
B)measures the proportion of variation in Y that is explained by X₁ and X_2.
C)measures the proportion of variation in Y that is explained by X₁ holding X₂ constant.
D)has the same sign as b₁.

Question 16

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
In examining the determinants of income,data were collected regarding the characteristics of 45 adults,and the regression lnY = β₀ + β₁ lnX₁ + β₂ lnX₂ + β₃X₃ + ε was used,where Y is the annual income (in thousands of dollars),X₁ is the adult's age,X₂ is his/her years of education,and X₃ is a dummy
variable = 1 and is used if the adult is female.You run the regression and obtain the equation
ln
= 6.3 + 0.91 lnx₁ + 1.3 ln x₂ - 0.05x₃.
How would you interpret the coefficient on age?

A)As the person's age increases,their income increases at a decreasing rate.
B)As the person's age increases,their income increases at an increasing rate.
C)As the person's age increases,their income at first increases,then eventually decreases.
D)As the person's age increases,their income at first decreases,then eventually increases.

Question 17

In testing the validity of a multiple regression model,a large value of the F-test statistic indicates that:

A)most of the variation in the independent variables is explained by the variation in y.
B)the model has significant explanatory power because at least one slope coefficient is not zero.
C)most of the variation in y is unexplained by the regression equation.
D)the model provides a poor fit.

Question 18

What is the approximate p-value associated with the hypothesis test for β₂?

A)0)027
B)0)05
C)0)10
D)0)25

Question 19

If a multiple regression analysis is based on five independent variables collected from a sample of 100 observations,what will be the value of the denominator in the calculation of the multiple standard error of estimate?

A)98
B)96
C)94
D)92

Question 20

How would you interpret the coefficient on x₂?

A)The average square footage area of living space is 1,250 square feet.
B)We would expect to pay approximately an additional $1,000 for each additional square foot of living space.
C)We would expect to pay an additional $1,000 for each an additional 125 square foot of living space,given that the number of bedrooms and the number of miles from the beach are held constant.
D)We would expect to pay,on average,an additional $125 for each additional square foot of living space,given that the number of bedrooms and the number of miles from the beach are held constant.

Question 21

What would we expect the total dollar value of loans to be in a month where there are 42 loans,the interest rate is 7.5%,and the bank spends $30,000 in advertising?

A)$6.442 million
B)$6.558 million
C)$6.288 million
D)$6.112 million

Question 22

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.The officer used Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,and X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars).Using data from the past 24 months,she obtained the following results:
= 5.7 + 0.189x₁ - 1.3x₂ + 0.08x₃,
= 3.2,
= 0.03,
= 0.062,
= 0.17,R² = 0.46,and adjusted
= 0.41.
What should the null and alternative hypotheses be for β₁?

A)H₀ : β₁ = 0, H₁ : β₁ > 0
B)H₀ : β₁ = 0,H₁ : β₁ < 0
C)H₀ : β₁ ≠ 0,H₁ : β₁ = 0
D)H₀ : β₁ > 0, H₁ : β₁ < 0

Question 23

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.She used Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄


+ ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,and X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars).Suppose that by using data from the past 24 months,she obtained
= 3.8 + 0.23x₁ - 1.31x₂ + 0.032x₃ - 0.0005
.
In multiple regression models,the values of the error variable ε are assumed to be:

A)autocorrelated.
B)dependent of each other.
C)independent of each other.
D)always positive.

Question 24

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.The officer used Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,and X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars).Using data from the past 24 months,she obtained the following results:
= 5.7 + 0.189x₁ - 1.3x₂ + 0.08x₃,
= 3.2,
= 0.03,
= 0.062,
= 0.17,R² = 0.46,and adjusted
= 0.41.
What should the null and alternative hypotheses be for β₃?

A)H₀ : β₃ = 0, H₁ : β₃ > 0
B)H₀ : β₃ = 0, H₁ : β₃ < 0
C)H₀ : β₃ ≠ 0, H₁ : β₃ = 0
D)H₀ : β₃ > 0, H₁ : β₃ ≠ 0

Question 25

In a multiple regression analysis involving 25 data points,the square of the standard error of the estimate is 1.9 and the sum of squares for error SSE is 38.Then,the number of the independent variables must be:

A)4
B)5
C)6
D)3

Question 26

What is the most accurate conclusion regarding the hypothesis test for β₃?

A)reject H₀ at α = 0.10
B)reject H₀ at α = 0.05
C)fail to reject H₀ at α ≤ 0.10
D)reject H₀ at α = 0.01

Question 27

What is the most accurate conclusion regarding the hypothesis test for β₂?

A)fail to reject H₀ at α > 0.10
B)reject H₀ at all levels of α
C)reject H₀₀ only at α = 0.05
D)fail to reject H₀ at α = 0.005

Question 28

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.The officer used lnY = β₀ + β₁lnX₁ + β₂lnX₂ + β₃lnX₃ + lnε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,and X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars).Using data from the past 24 months,she obtained ln
= 0.67 + 1.2 ln x₁ - 1.45 ln x₂ + 1.07 ln x₃.
How would the officer interpret the coefficient on x₂?

A)As the interest rate increases,the total loan amount decreases at a decreasing rate.
B)As the total loan amount increases,the interest rate decreases at an increasing rate.
C)As the interest rate increases,the total loan amount decreases at an increasing rate.
D)As the total loan amount increases,the interest rate decreases at a decreasing rate.

Question 29

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.The officer used Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,and X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars).Using data from the past 24 months,she obtained the following results:
= 5.7 + 0.189x₁ - 1.3x₂ + 0.08x₃,
= 3.2,
= 0.03,
= 0.062,
= 0.17,R² = 0.46,and adjusted
= 0.41.
What should the null and alternative hypotheses be for β₂?

A)H₀ : β₂ ≠ 0, H₁ : β₂ > 0
B)H₀ : β₂ = 0, H₁ : β₂ < 0
C)H₀ : β₂ ≠ 0, H₁ : β₂ = 0
D)H₀ : β₂ > 0, H₁ : β₂ ≠ 0

Question 30

In a multiple regression model,the mean of the probability distribution of the error variable,ε_i,is assumed to be:

A)1.0
B)0.0
C)any value greater than 1.
D)K,where K is the number of independent variables included in the model.

Question 31

How would we interpret the coefficient of determination R²?

A)Approximately 46% of the time,the dollar value of the loans is determined by these three variables.
B)Approximately 46% of the observations lie within 1 standard deviation of the regression line.
C)Approximately 46% of the variation in the total dollar amount of loans can be explained by the variation in these three variables.
D)Approximately 46% of the total dollar amount of loans is explained by these three variables.

Question 32

How should the loan officer interpret the coefficient on x₃?

A)For every additional $8,000 spent on advertising,we would expect the total dollar value of residential loans to increase by $1,000,000 on average,assuming that all the other independent variables in the model are held constant.
B)For every additional dollar spent on advertising,we would expect the total dollar value of residential loans to increase by $0.08 million,assuming that all the other independent variables in the model are held constant.
C)For every additional $80,000 spent on advertising,we would expect the total dollar value of residential loans to increase by $1,000,000,assuming that all the other independent variables in the model are held constant.
D)For every additional $1000 spent on advertising,we would expect the total dollar value of residential loans to increase by $80,000 on average,assuming that all the other independent variables in the model are held constant.

Question 33

What is the 95% confidence interval associated with β₁?

A)0)016 ± 0.189
B)0)189 ± 0.016
C)0)189 ± 0.026
D)0)189 ± 0.063

Question 34

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.The officer used lnY = β₀ + β₁lnX₁ + β₂lnX₂ + β₃lnX₃ + lnε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,and X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars).Using data from the past 24 months,she obtained ln
= 0.67 + 1.2 ln x₁ - 1.45 ln x₂ + 1.07 ln x₃.
How would the officer interpret the coefficient on x₃?

A)As the total loan amount increases,the amount of advertising increases at an increasing rate.
B)As the amount of advertising increases,the total loan amount increases at an increasing rate.
C)As the amount of advertising increases,the total loan amount increases at a decreasing rate.
D)As the total loan amount increases,the amount of advertising increases at a decreasing rate.

Question 35

What is the 95% confidence interval for β₁?

A)2)13 ± 4.17
B)4)59 ± 4.38
C)2)13 ± 4.38
D)4)59 ± 4.17

Question 36

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.She used Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄


+ ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,and X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars).Suppose that by using data from the past 24 months,she obtained
= 3.8 + 0.23x₁ - 1.31x₂ + 0.032x₃ - 0.0005
.
What do these results suggest about the relationship between the total loan amount and advertising?

A)As the amount of advertising increases,the total loan amount decreases at a decreasing rate.
B)As the amount of advertising increases,the total loan amount at first decreases,then increases.
C)As the amount of advertising increases,the total loan amount increases at an increasing rate.
D)As the amount of advertising increases,the total loan amount at first increases,then decreases.

Question 37

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.She used Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄


+ ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,and X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars).Suppose that by using data from the past 24 months,she obtained
= 3.8 + 0.23x₁ - 1.31x₂ + 0.032x₃ - 0.0005
.
What do these results suggest about the relationship between the total loan amount and number of loans?

A)As the number of loans increases,the total loan amount at first decreases,then increases.
B)As the number of loans increases,the total loan amount also increases.
C)As the number of loans increases,the total loan amount at first increases,then decreases.
D)As the number of loans increases,the total loan amount decreases at a decreasing rate.

Question 38

In order to test the validity of a multiple regression model involving 4 independent variables and 25 observations,the numerator and denominator degrees of freedom for the critical value of F are ________ respectively.

A)4 and 20
B)4 and 25
C)5 and 24
D)5 and 25

Question 39

In multiple regression analysis,the ratio MSR/ yields the:

A)F-test statistic for testing the validity of the regression equation.
B)t-test statistic for testing each individual regression coefficient.
C)multiple coefficient of determination.
D)adjusted multiple coefficient of determination.

Question 40

How should the loan officer interpret the coefficient on x₂?

A)For an additional 1.3 percent increase in the interest rate,we would expect the total dollar value of residential loans to decrease by $1.0 million,assuming that all the other independent variables in the model are held constant.
B)For an additional percent increase in the interest rate,we would expect the total dollar value of residential loans to decrease by $1.3 million on average,assuming that all the other independent variables in the model are held constant.
C)For an additional million dollars in loans,we would expect the interest rate to decrease by 1.3 percent,assuming that all the other independent variables in the model are held constant.
D)For an additional $1.3 million in loans,we would expect the interest rate to decrease by 1.0 percent on average,assuming that all the other independent variables in the model are held constant.

Question 41

In a multiple regression model,the following statistics are given: SSE = 100,R² = 0.995,K = 5,and n = 15.Determine the multiple coefficient of determination adjusted for degrees of freedom.

A)0.955
B)0.992
C)0.900
D)0.855

Question 42

To test the validity of a multiple regression model,we test the null hypothesis that the regression coefficients are all zero by applying the:

A)F-test
B)t-test
C)z-test
D)chi-square test

Question 43

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.She used Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars),and X₄ is a dummy variable equal to 1 if the observation is either June,July,or August.
Suppose that she obtained = 3.8 + 0.23x₁ - 1.31x₂ + 0.032x₃ + 1.05x₄ - 0.22x₃x₄ by using data from the past 24 months.How would we interpret the coefficient of x₃x₄?

A)On an average,we would expect $220,000 less in loans during the months of June,July,and August.
B)Advertising is more effective during June,July,and August.
C)Advertising is less effective in June,July,and August.
D)On an average,we would expect $220,000 more in loans during the months of June,July,and August.

Question 44

In a regression model involving 50 observations,the following estimated regression model was obtained: = 8.6 + 2.3x₁ + 3.8x₂ + 6.1x₃.For this model,the following statistics are given: SSR = 360 and SSE = 129.The value of MSR is:

A)120
B)163
C)90
D)43

Question 45

On an average,how much would we expect a 5-member household with a monthly income of $2,100 to spend on groceries? Assume that only one of the adults is working.

A)$215.30
B)$481.60
C)$172.10
D)$438.40

Question 46

In a multiple regression analysis involving K independent variables and n data points,the number of degrees of freedom associated with the sum of squares for error is:

A)K-1
B)n - K
C)n - 1
D)n - K - 1

Question 47

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan officer is interested in examining the determinants of the total dollar value of residential loans made during a month.She used Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),X₁ is the number of loans,X₂ is the interest rate,X₃ is the dollar value of expenditures of the bank on advertising (in thousands of dollars),and X₄ is a dummy variable equal to 1 if the observation is either June,July,or August.
Suppose that she obtained = 3.8 + 0.23x₁ - 1.31x₂ + 0.032x₃ + 1.05x₄ by using data from the past 24 months.How would we interpret the coefficient on x₄?

A)On an average,we would expect an additional 1.05 loans per month during June,July,and August.
B)On an average,we would expect an additional $1.05 million in loans during June,July,and August.
C)On an average,we would expect an additional 1.05% increase in loans during June,July,and August.
D)There is no difference in the total residential loan amount between the months of June and September.

Question 48

A multiple regression model has the form = 8 + 3x₁ + 5x₂ 4x₃.As x₃ increases by one unit,with x₁ and x₂ held constant,then y,on average,is expected to:

A)increase by 1 unit.
B)decrease by 4 units.
C)increase by 12 units.
D)decrease by 16 units.

Question 49

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A researcher is interested in determining the monthly household expenditures on groceries.He models the relationship as follows: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε,where Y is the monthly dollar value spent on groceries,X₁ is the household's monthly income,X₂ is the number of people living in the household,and X₃ is a dummy variable = 1 if both the adults in the household work.Taking a sample of 250 households,the researcher gets the following results:
= 172.1 + 0.048x₁ + 33.1x₂ + 43.2x₃.
How should the researcher interpret the coefficient of the dummy variable?

A)Households with both adults working approximately spend $43.20 on groceries,under the assumption that all the other independent variables in the model are held constant.
B)Households with both adults working spend,on average,$43.20 less than other households,under the assumption that all the other independent variables in the model are held constant.
C)Households with both adults working spend,on average,$43.20 more than other households,under the assumption that all the other independent variables in the model are held constant.
D)There is no difference between the types of households and their spending on groceries,under the assumption that all the other independent variables in the model are held constant.

Question 50

A multiple regression model involves 5 independent variables and a sample of 10 data points.If we want to test the validity of the model at the 5% significance level,the critical value is:

A)6.26
B)3.33
C)9.36
D)4.24

Question 51

In a multiple regression problem involving two independent variables X₁ and X₂,what does it mean if the value of b₁ is 3.2?

A)The relationship between X₁ and Y is significant.
B)The estimated value of Y increases by an average of 3.2 units for each unit increase of X₁,holding X₂ constant.
C)The estimated value of Y increases approximately by 3.2 units for each unit increase of X₂,holding X₁ constant.
D)The estimated average value of Y is 3.2 units when X₁ = 0.

Question 52

In multiple regression analysis involving 10 independent variables and 80 observations,the critical value of t for testing individual coefficients in the model will have:

A)70 degrees of freedom.
B)10 degrees of freedom.
C)69 degrees of freedom.
D)9 degrees of freedom.

Question 53

What role does the adjusted coefficient of determination, ²,play in multiple regression analysis?

A)It tests the significance of all independent variables.
B)It corrects the coefficient R² for the number of independent variables.
C)It helps detect nonsignificant predictor variables.
D)It helps in the determination of the F-statistic for evaluating the overall statistical significance of the model.

Question 54

What is the range of values for the multiple coefficient of determination?

A)-1 to 1 inclusive
B)-1 to 0 inclusive
C)0 to 1 inclusive
D)-1 to 1 exclusive

Question 55

How many degrees of freedom are associated with a multiple regression model involving K predictors and n observations when running a t-test for the individual coefficients?

A)n - K
B)n - K - 1
C)n - K + 1
D)K - 1

Question 56

To test the validity of a multiple regression model involving three independent variables,the null hypothesis states that:

A)β₀ = β₁ = β₂ = β₃
B)β₁ = β₂ = β₃ = 0
C)β₁ = β₂ = β₃
D)β₁ ≠ β₂ ≠ β₃

Question 57

A multiple regression analysis involving three independent variables and 25 data points results in a value of 0.769 for the unadjusted multiple coefficient of determination.The adjusted multiple coefficient of determination is:

A)0.385
B)0.877
C)0.591
D)0.736

Question 58

Which of the following formulas would you use to calculate the multiple coefficient of determination?

A)
B)
C)
D)

Question 59

If we run the regression Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε using a sample of 30 observations,what is the alternative hypothesis for the F-test of overall significance?

A)H₁ : at least one β_j ≠ 0
B)H₁ : β₁ = β₂ = β₃x₃ = β₄ = 0
C)H₁ : β₀ = β₁ = β₂ = β₃x₃ = β₄ = 0
D)H₁ : β₁ ≠ β₂ ≠ β₃x₃ ≠ β₄ ≠ 0

Question 60

A multiple regression model involves 10 independent variables and 30 observations.If we want to test at the 5% significance level the parameter β₄,the critical value will be:

A)1.697
B)2.093
C)2.228
D)1.729

Question 61

The multiple coefficient of determination for a multiple regression model with two explanatory variables X₁ and X₂:

A)measures the variation around the predicted regression equation.
B)measures the proportion of variation in Y that is explained by X₁ and X₂.
C)measures the proportion of variation in Y that is explained by X₁ holding X₂ constant.
D)has the same sign as b₁.

Question 62

The two regressions Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε and Y = β₀ + β₁X₁ + β₂X₂ + ε were run using a sample of 60 observations.If the SSE for the first regression is 1,688.4,what would you expect the SSE for the second regression be?

A)SSE = 1,688.4
B)SSE < 1,688.4
C)SSE ≥ 1,688.4
D)SSE < 688.4

Question 63

Consider the F-test of overall significance.If we reject H₀ : β₁ = β₂ = β₃ = β₄ = 0,what can you conclude?

A)At least one of the independent variables is statistically significant in explaining the variation in the dependent variable.
B)All of the independent variables are statistically significant in explaining the variation in the dependent variable.
C)The model is not statistically significant in explaining the variation in the dependent variable.
D)The model is a good predictor of the dependent variable.

Question 64

In a multiple regression model,there are six independent variables and 120 observations.What are the degrees of freedom associated with the mean square regression?

A)120
B)6
C)119
D)5

Question 65

In a regression model involving 30 observations,the following estimated regression model was obtained: = 5.1 + 1.5x₁ + 1.2x₂.For this model,the following statistics were given: SST = 162.1 and SSE = 73.9.What is the value of the F statistic for testing the validity of this model?

A)5)49
B)16.11
C)44.10
D)2)737

Question 66

What can the manager infer about the relationship between the square of the level of output and the average costs?

A)As the level of output increases,average costs decrease at an increasing rate.
B)As the level of output increases,at first average costs decrease,then will start to increase.
C)As the level of output increases,at first average costs increase,then will start to decrease.
D)As the level of output increases,average costs increase at an increasing rate.

Question 67

The manager's null and alternative hypotheses for β₁ is:

A)H₀ : β₁ = 0 and H₁ : β₁ > 0.
B)H₀ : β₁ > 0 and H₁ : β₁ < 0.
C)H₀ : β₁ ≠ 0 and H₁ : β₁ = 0.
D)H₀ : β₁ = 0 and H₁ : 0 < β₁ < 1.

Question 68

In a regression model involving 34 observations,the following estimated regression model was obtained: = 48 + 2.5x₁ + 1.2x₂ - 0.7x₃.For this model,the following statistics were given: SST = 960 and SSE = 270.What is the value of the F statistic for testing the validity of this model?

A)25.56
B)7.94
C)28.24
D)22.26

Question 69

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
The data set shown below gives the information to model a regression that predicts the percent profit margin.
Y = annual profit margin
X₁ = Year
X₂ = net annual revenue per deposit dollar
X₃ = number of savings and loan offices for that year

What is the value of b₀?

A)1.2432
B)0.2432
C)0.2003
D)0.8514

Question 70

In a multiple regression model,which of the following is true of the value of R² adjusted for the degrees of freedom?

A)It can be negative.
B)It has to be positive.
C)It has to be larger than the multiple coefficient of determination.
D)It can be larger than 1.

Question 71

In a multiple regression model,there are four independent variables and 60 observations.What are the degrees of freedom associated with the error sum of squares?

A)56
B)4
C)55
D)3

Question 72

In a multiple regression analysis,six independent variables are used in the equation based on a sample of 45 observations.What are the degrees of freedom associated with the F statistic?

A)6 and 45
B)5 and 44
C)6 and 38
D)4 and 40

Question 73

In a multiple regression analysis,which of the following is used for testing hypotheses concerning individual regression coefficients?

A)t-statistic
B)z-statistic
C)F statistic
D)chi-square test

Question 74

In a multiple regression model,there are two independent variables and 25 observations.If SSE = 0.0625 and SST = 0.475,what is the value of the adjusted coefficient of determination?

A)0.8684
B)0.1435
C)0.9802
D)0.8565

Question 75

Most statistical softwares provide the p-value for testing each coefficient in the multiple regression model.In the case of the regression coefficient b₃,this p-value represents the probability that:

A)b₃ = 0.
B)β₃ = 0.
C)
could be this large if β₃ = 0.
D)
could be this large if β₃ ≠ 0.

Question 76

A multiple regression model has the form: = 3.25 + 2.5x₁ + 1.5x₂.If x₂ increases by 1 unit,holding x₁ constant,the value of y will increase on average by:

A)3.25 units.
B)2.5 units.
C)1.5 units.
D)7.25 units.

Question 77

If an additional variable,however irrelevant,is added to a multiple regression model,what is the most likely result?

A)a smaller sum of squared errors
B)a larger sum of squared errors
C)a constant sum of squared errors
D)a smaller regression sum of squares

Question 78

Calculate the value of b₁.

A)0.2800
B)1.8000
C)0.0169
D)0.0201

Question 79

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A production manager is interested in modeling the determinants of the average cost of production.He creates the following model: Y = β₀ + β₁X₁ + β₂X₂ + β₃


+ ε,where X₁ is the cost per unit of the primary input,and X₂ is the level of output.He examines the records over the past 45 production runs and obtains the following results:
.
What would the manager's null and alternative hypotheses for testing the significance of β₂ be?

A)H₀ : β₂ = 0 and H₁ : β₂ > 0
B)H₀ : β₂ > 0 and H₁ : β₂ < 0
C)H₀ : β₂ ≠ 0 and H₁ : β₂ = 0
D)H₀ : β₂ = 0 and H₁ : 0 < β₂ < 1

Question 80

In a multiple regression problem involving two independent variables X₁ and X₂,what does it mean if b₂ is computed to be -1.5?

A)The relationship between X₂ and Y is significant.
B)The estimated value of Y decreases by an average of 1.5 units for each unit increase of X₁,holding X₂ constant.
C)The estimated value of Y decreases by an average of 1.5 units for each unit increase of X₂,holding X₁ constant.
D)The estimated average value of Y is -1.5 units when X₁ = X₂ = 0.