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 = βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + ε where Y = price of the house in $1,000s,XS1U1B11S1U1B0 = number of bedrooms,XS1U1B12S1U1B0S1U1B1 S1U1B0= square footage of living space,and XS1U1B13S1U1B0S1U1B1 S1U1B0= 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 + 4.59xS1U1B11S1U1B0 + 0.125xS1U1B12S1U1B0 - 6.04xS1U1B13S1U1B0, = 103.2, = 2.13, = 0.062, = 4.17,RS1U1P12S1S1P0 = 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

Accepted Answer

A

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

Accepted Answer

The price of a house is influenced by various factors including location, size, and proximity to amenities like the beach. Given the details, $243,850 is a reasonable estimate for a 3-bedroom house near the beach.

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 = βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 +ε was used,where Y is the annual income (in thousands of dollars),XS1U1B11S1U1B0 is the person's age,XS1U1B12S1U1B0 is his/her years of education,and XS1U1B13S1U1B0 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.

Accepted Answer

A

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 = βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + ε where Y = price of the house in $1,000s,XS1U1B11S1U1B0 = number of bedrooms,XS1U1B12S1U1B0S1U1B1 S1U1B0= square footage of living space,and XS1U1B13S1U1B0S1U1B1 S1U1B0= 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 + 4.59xS1U1B11S1U1B0 + 0.125xS1U1B12S1U1B0 - 6.04xS1U1B13S1U1B0, = 103.2, = 2.13, = 0.062, = 4.17,RS1U1P12S1S1P0 = 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

Accepted Answer

The null hypothesis (H0) states that the coefficient β1 is equal to 0, while the alternative hypothesis (H1) states that β1 is greater than 0. This is because we are interested in testing whether the independent variable has a positive effect on the dependent variable.

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

Accepted Answer

The hypothesis test concludes that we reject H₀ at α = 0.10, indicating sufficient evidence against the null hypothesis at this significance level.

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 X₂is the person's age.What would be your H₀regarding 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

Accepted Answer

The regression equation includes two coefficients for the age variable, β₂ and β₃, which suggest a non-linear effect of age on wealth. In particular, β₂ represents the effect of age on wealth when age is positive, while β₃ captures the effect of age on wealth when age is negative. Since the question assumes that age has a non-linear effect on wealth, we can expect that the two coefficients will have opposite signs. Specifically, β₂ should be positive (indicating that wealth increases with age for some range of ages) while β₃ should be negative (indicating that wealth decreases with age for some range of ages). Therefore, the best choice is C, which states that 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

Accepted Answer

Dummy variables are created when independent variables are categorical in multiple regression analysis. Dummy variables help to represent categorical variables in a numerical format, where each variable is coded as a 0 or 1. This allows the model to assess the unique contribution of each category to the dependent variable. Therefore, dummy variables are the best choice for applying multiple regression to situations when independent variables are categorical.

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.

Accepted Answer

If we fail to reject the null hypothesis, it means that we do not have sufficient evidence to conclude that there is a relationship between the dependent variable and the independent variables. Therefore, choice A is the best option. The other options are not necessarily supported by failing to reject the null hypothesis.

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.

Accepted Answer

The coefficient of determination (R-squared) measures the proportion of variability in the dependent variable (sales prices in this case) that can be explained by the independent variables (R² in this case). Therefore, choice B is the correct interpretation, which states that 47% of the variation in sales prices can be determined by the variation in these three variables. The other choices are incorrect interpretations of the coefficient of determination.

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 = βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 +ε was used,where Y is the annual income (in thousands of dollars),XS1U1B11S1U1B0 is the person's age,XS1U1B12S1U1B0 is his/her years of education,and XS1U1B13S1U1B0 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.

Accepted Answer

The coefficient on gender in the regression equation is -0.76, which means that on average, a woman earns $760 less than a man, taking into account age and education.

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 = βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + ε where Y = price of the house in $1,000s,XS1U1B11S1U1B0 = number of bedrooms,XS1U1B12S1U1B0S1U1B1 S1U1B0= square footage of living space,and XS1U1B13S1U1B0S1U1B1 S1U1B0= 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 + 4.59xS1U1B11S1U1B0 + 0.125xS1U1B12S1U1B0 - 6.04xS1U1B13S1U1B0, = 103.2, = 2.13, = 0.062, = 4.17,RS1U1P12S1S1P0 = 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

Accepted Answer

The null hypothesis should be that β₃= 0 (meaning that the number of miles from the beach does not significantly affect the price of the house), and the alternative hypothesis should be that β₃< 0 (meaning that the number of miles from the beach has a negative effect on the price of the house). This is because a lower number of miles from the beach is likely to be seen as more desirable, and therefore have a positive effect on the price, while a higher number of miles from the beach is likely to be seen as less desirable, and therefore have a negative effect on the price.

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 = βS1U1B10S1U1B0S1U1B1 S1U1B0+ βS1U1B11S1U1B0S1U1B1 S1U1B0lnXS1U1B11S1U1B0 + βS1U1B12S1U1B0 lnXS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + ε was used,where Y is the annual income (in thousands of dollars),XS1U1B11S1U1B0 is the adult's age,XS1U1B12S1U1B0 is his/her years of education,and XS1U1B13S1U1B0 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 lnxS1U1B11S1U1B0 + 1.3 ln xS1U1B12S1U1B0 - 0.05xS1U1B13S1U1B0.
-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.

Accepted Answer

The answer of THE NEXT QUESTIONS ARE BASED ON THE...

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.

Accepted Answer

The coefficient on x₃ indicates the effect of increasing the distance from the beach by one mile on the housing price, holding the number of bedrooms and square footage of living space constant. The coefficient value is negative, -6040, which implies that we would expect to pay less for a house that is located an additional mile away from the beach. Therefore, the best choice is D.

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

Accepted Answer

The reduction in SSE from 382.3 to 298.4 when adding the variables β13 and β14 suggests that these variables significantly improve the model fit, indicating that the null hypothesis (that these coefficients are equal to zero) should be rejected at a common significance level of α = 0.05.

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 X₂_. C)measures the proportion of variation in Y that is explained by X₁ holding X₂ constant. D)has the same sign as b₁.

Accepted Answer

The standard error of the estimate measures the variation around the predicted regression equation, indicating the degree of scatter around the regression line. It gives an estimate of the average amount by which the observed Y values differ from the predicted Y values based on the regression equation. It is not related to the proportion of variation explained by the explanatory variables or the regression coefficients. The sign of the regression coefficients (b) reflects the direction of the relationship between the explanatory variables and the response variable, but the standard error of the estimate is not directly related to it.

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 = βS1U1B10S1U1B0S1U1B1 S1U1B0+ βS1U1B11S1U1B0S1U1B1 S1U1B0lnXS1U1B11S1U1B0 + βS1U1B12S1U1B0 lnXS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + ε was used,where Y is the annual income (in thousands of dollars),XS1U1B11S1U1B0 is the adult's age,XS1U1B12S1U1B0 is his/her years of education,and XS1U1B13S1U1B0 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 lnxS1U1B11S1U1B0 + 1.3 ln xS1U1B12S1U1B0 - 0.05xS1U1B13S1U1B0.
-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.

Accepted Answer

The coefficient on age is 0.91, which indicates that as the person's age increases, their income increases. However, the coefficient is less than 1, which means that the increase in income is at a decreasing rate. This suggests that while older individuals may have more experience and skills, they may also face age discrimination and/or limited job opportunities.

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.

Accepted Answer

A large value of the F-test statistic indicates that at least one of the independent variables has a significant explanatory power in predicting the dependent variable. This means that the model has significant explanatory power because at least one slope coefficient is not zero. Choice A is incorrect because a high F-test statistic indicates that much of the variation in y is explained, not in the independent variables. Choice C is incorrect because a high F-test statistic indicates that the model explains a significant amount of variation in y. Choice D is incorrect because a high F-test statistic indicates a good fit for the model.

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

Accepted Answer

Without additional information provided, we cannot determine the exact p-value. However, typically a p-value less than 0.05 is considered statistically significant and sufficient evidence to reject the null hypothesis. The closest option to this is choice A, 0.027, which is less than 0.05. Therefore, A is the best choice.

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

Accepted Answer

The denominator in the calculation of the multiple standard error of estimate is the degrees of freedom, which is calculated as the number of observations minus the number of parameters estimated (including the intercept). In this case, it is 100 - (5 + 1) = 94.

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.

Accepted Answer

The coefficient on a variable in a regression model typically represents the expected change in the dependent variable (e.g., price) for a one-unit increase in the independent variable (e.g., square footage), holding all other variables constant. Therefore, choice D correctly interprets the coefficient as the expected additional cost per square foot, assuming other factors like the number of bedrooms and distance from the beach are constant.

Quiz 12: Multiple Regression

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.

What is your conclusion regarding the hypothesis test for β3?

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

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

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

How would you interpret the coefficient on x3?

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.

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

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

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

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?

How would you interpret the coefficient on x2?

What is your conclusion regarding the hypothesis test for β₃?

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

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

How would you interpret the coefficient on x₃?

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.

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

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

How would you interpret the coefficient on x₂?