Question 1

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

Accepted Answer

The formula for the multiple coefficient of determination adjusted for degrees of freedom is: Adjusted R-square = 1 - (SSE/(n-k-1))/(SST/(n-1)) where SSE is the sum of squared errors, n is the sample size, k is the number of independent variables, and SST is the total sum of squares. Substituting the given values, we get: Adjusted R-square = 1 - (100/(15-5-1))/(R²*(15-1)) = 0.992 Therefore, the best choice is B.

Question 2

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

Accepted Answer

The F-test is the appropriate test to determine the validity of a multiple regression model by evaluating the null hypothesis that all regression coefficients are equal to zero. The F-test compares the ratio of the explained variance to the unexplained variance in a model, and if the ratio is significant, it indicates that the model is a good fit and the null hypothesis is rejected. Therefore, the F-test is the best choice to test the validity of a multiple regression model.

Question 3

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 = βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + βS1U1B14S1U1B0XS1U1B14S1U1B0 + ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),XS1U1B11S1U1B0S1U1B1 S1U1B0is the number of loans,XS1U1B12S1U1B0S1U1B1 S1U1B0is the interest rate,XS1U1B13S1U1B0 is the dollar value of expenditures of the bank on advertising (in thousands of dollars),and XS1U1B14S1U1B0 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.

Accepted Answer

The coefficient of x₃x₄ is -0.22, which means that the effect of advertising (X₃) on the total dollar value of loans (Y) is lower during the months of June, July, and August (X₄ = 1), compared to other months (X₄ = 0). Therefore, advertising is less effective during those months.

Question 4

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

Accepted Answer

The answer of In a regression model involving 50 observations,the...

Question 5

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

Accepted Answer

The answer of On an average,how much would we expect...

Question 6

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

Accepted Answer

The number of degrees of freedom associated with the sum of squares for error is equal to the total number of observations (n) minus the number of parameters being estimated, which is K + 1 (one for the intercept and K for the independent variables). Therefore, the correct formula for the degrees of freedom is n - (K + 1), which simplifies to n - K - 1. Option D is the only choice that matches this formula.

Question 7

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 = βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + βS1U1B14S1U1B0XS1U1B14S1U1B0 + ε to model the relationship,where Y is the total dollar value of residential loans in a month (in millions of dollars),XS1U1B11S1U1B0S1U1B1 S1U1B0is the number of loans,XS1U1B12S1U1B0S1U1B1 S1U1B0is the interest rate,XS1U1B13S1U1B0 is the dollar value of expenditures of the bank on advertising (in thousands of dollars),and XS1U1B14S1U1B0 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.

Accepted Answer

The coefficient for the dummy variable x₄ is 1.05, which means that on an average, we would expect an additional $1.05 million in residential loans during the months of June, July, and August compared to the other months. Therefore, option B is the correct interpretation of the coefficient.

Question 8

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.

Accepted Answer

The answer of A multiple regression model has the form...

Question 9

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 = βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + ε,where Y is the monthly dollar value spent on groceries,XS1U1B11S1U1B0 is the household's monthly income,XS1U1B12S1U1B0 is the number of people living in the household,and XS1U1B13S1U1B0S1U1B1 S1U1B0is 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 + 0.048xS1U1B11S1U1B0 + 33.1xS1U1B12S1U1B0 + 43.2xS1U1B13S1U1B0.
-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.

Accepted Answer

The coefficient of a dummy variable in a regression model represents the average difference in the dependent variable (in this case, spending on groceries) for the group represented by the dummy variable (households with both adults working) compared to the reference group (households with not both adults working), holding all other variables constant. Therefore, if the coefficient is positive (as indicated by the $43.20), it means that, on average, households with both adults working spend $43.20 more on groceries than other households, assuming all other factors are equal.

Question 10

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

Accepted Answer

The answer of A multiple regression model involves 5 independent...

Question 11

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.

Accepted Answer

The value of b₁ being 3.2 means that for each unit increase in X₁, while holding X₂ constant, the estimated value of Y increases by an average of 3.2 units. Therefore, option B is the correct choice. Option A is incorrect as we do not know if the relationship is significant based on the coefficient value alone. Option C is incorrect as it incorrectly states the relationship between X₁ and Y. Option D is incorrect as it assumes a specific value for X₁ and does not account for other possible values.

Question 12

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.

Accepted Answer

In multiple regression analysis, the degrees of freedom for testing individual coefficients is calculated as the number of observations minus the number of parameters being estimated (including the intercept). Here, with 80 observations and 10 independent variables plus 1 intercept, the degrees of freedom = 80 - (10 + 1) = 69.

Question 13

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.

Accepted Answer

The answer of What role does the adjusted coefficient of...

Question 14

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

Accepted Answer

The multiple coefficient of determination (R-squared) ranges from 0 to 1 inclusive, as it represents the proportion of variance in the dependent variable that is explained by the independent variables in a multiple regression model. The values cannot be negative since it represents the squared correlation between the predicted and actual values of the dependent variable. Therefore, options A, B, and D are incorrect.

Question 15

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

Accepted Answer

When running a t-test for the individual coefficients in a multiple regression model involving K predictors and n observations, the degrees of freedom are equal to n - K - 1. This is because one degree of freedom is lost for each predictor in the model, and an additional degree of freedom is lost for the intercept term. Therefore, the best choice is (B) n - K - 1.

Question 16

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

Accepted Answer

The null hypothesis for testing the validity of a multiple regression model involving three independent variables states that all the regression coefficients, except the intercept, are equal to zero. Therefore, the correct choice is B, which states that all the regression coefficients for the three independent variables are equal to zero.

Question 17

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

Accepted Answer

The adjusted R-squared accounts for the number of predictors in the model and the sample size. It is calculated using the formula: Adjusted R² = 1 - [(1-R²)(n-1)/(n-k-1)], where R² is the unadjusted R-squared, n is the number of observations, and k is the number of predictors. Plugging in the values: Adjusted R² = 1 - [(1-0.769)(25-1)/(25-3-1)] = 0.736.

Question 18

Which of the following formulas would you use to calculate the multiple coefficient of determination?
A) 
B) 
C) 
D)

Accepted Answer

The answer of Which of the following formulas would you...

Question 19

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

Accepted Answer

The F-test of overall significance tests whether at least one of the coefficients in the model is significantly different from zero. The alternative hypothesis for this test is that at least one of the coefficients in the model is not equal to zero. The correct choice is A, as it presents an alternative hypothesis that includes all the coefficients (β₁, β₂, β₃, β₄) in the model.

Question 20

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

Accepted Answer

The degrees of freedom for the t-distribution is n - k - 1, where n is the number of observations and k is the number of independent variables. Here, df = 30 - 10 - 1 = 19. At a 5% significance level for a two-tailed test, the critical t-value for 19 degrees of freedom is approximately 2.093.

Quiz 12: Multiple Regression

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.

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

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:

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.

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 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 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:

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

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:

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

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

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?

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

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:

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

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 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:

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.

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 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:

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

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

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 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: