Exam 12: Multiple Regression

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: The model y = β₀ + β₁X₁ + β₂X₂ + ε was fitted to a sample of 25 families in order to explain household milk consumption: where y = Milk consumption,in quarts,per week,x₁ = Weekly income,in hundreds of dollars,and x₂ = Family size.The least squares estimates of the regression parameters were b₀ = -0.03,b₁ = 0.05,and b₂ = 1.1,with coefficient standard errors = 0.02; = 0.38.The total sum of squares and the error sum of squares were found to be 165.8 and 66.32 respectively. -Find and interpret the coefficient of multiple correlation.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Suppose you are interested in determining the factors that influence the time required to prepare a tax return,and developed the model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε,where Y is the amount of time (in minutes),X₁ is the income of the individual (in thousands of dollars),X₂ is the age of the individual,X₃ is the number of people living in the household,and X₄ is a dummy variable that takes the value 1,if the individual owns his or her own home.After interviewing 40 accountants,you get the following results: = 17.2 + 3.8x₁ - 1.04x₂ + 2.15x₃ + 15.1x₄, = 5.3, = 0.13, = 0.33, = 1.51, = 4.7,SSR = 164.2,SSE = 200.7,and R² = 0.45. -What would you expect regarding these population regression parameters?

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?

Determine the value of .

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:

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: As director of the local tourist board,you are interested in determining the factors that influence the hotel occupancy rate in your city each month.Hotel occupancy can be measured as the percentage of available hotel rooms that are occupied by paying customers.You develop the following model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε,where Y is the hotel occupancy rate,X₁ is the total number of passengers arriving at the airport,X₂ is a price index of local hotel room rates,X₃ is the consumer confidence index,and X₄ is a dummy variable = 1 during the months of June,July,and August.You look at data from the past 36 months and obtain the following results: = 67.1 + 0.02x₁ - 0.055x₂ + 0.08x₃ + 12.3x₄,R² = 0.67, = 58.3, = 0.008, = 0.01, = 0.06, = 4.7,and SSE = 576. -Interpret the estimated regression coefficient b₁.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: In a study of foreign holdings in Egyptian banks,the following sample regression results were obtained,based on 14 annual observations: = -3.25 + - + ,and R² = 0.92, Where the numbers in parentheses under the coefficient estimates are the estimated coefficient standard errors,and y = Year-end share of assets in Egyptian bank subsidiaries held by foreigners,as a percentage of total assets x₁ = Annual change,in billions of Egyptian pounds,in foreign direct investment in Egypt x₂ = Bank price-earnings ratio x₃ = Index of the exchange value of the Egyptian pounds -Test the null hypothesis that β₁ is zero,against the alternative that it is positive at the 5% significance level,and interpret your result.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Assume that x₁ is a continuous variable and x₂ is a dummy variable with a value of 0 or 1. -What are the model constant and the slope coefficient of x₁ when x₂ equals 1 in the equation ?

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

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: The computer output for the multiple regression model,y = β₀ + β₁X₁ + β₂X₂ + ε is shown below.However,because of a printer malfunction some of the results are not shown.These are identified by asterisks. S = * R-Sq = * ANALYSIS OF VARIANCE -Calculate the error sum of squares.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Suppose you are interested in determining the factors that influence the time required to prepare a tax return,and developed the model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε,where Y is the amount of time (in minutes),X₁ is the income of the individual (in thousands of dollars),X₂ is the age of the individual,X₃ is the number of people living in the household,and X₄ is a dummy variable that takes the value 1,if the individual owns his or her own home.After interviewing 40 accountants,you get the following results: = 17.2 + 3.8x₁ - 1.04x₂ + 2.15x₃ + 15.1x₄, = 5.3, = 0.13, = 0.33, = 1.51, = 4.7,SSR = 164.2,SSE = 200.7,and R² = 0.45. -Interpret the estimated regression coefficient b₂.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: The model y = β₀ + β₁X₁ + β₂X₂ + ε was fitted to a sample of 25 families in order to explain household milk consumption: where y = Milk consumption,in quarts,per week,x₁ = Weekly income,in hundreds of dollars,and x₂ = Family size.The least squares estimates of the regression parameters were b₀ = -0.03,b₁ = 0.05,and b₂ = 1.1,with coefficient standard errors = 0.02; = 0.38.The total sum of squares and the error sum of squares were found to be 165.8 and 66.32 respectively. -Find and interpret the coefficient of determination.

The value of is:

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: The director of a local tourist board is interested in determining the factors that influence the hotel occupancy rate in his city each month.Hotel occupancy can be measured as the percentage of available hotel rooms that are occupied by paying customers.He develops the model: lnY = β₀ + β₁lnX₁ + β₂lnX₂ + β₃lnX₃ + β₄X₄ + ε,where Y is the hotel occupancy rate (as a percentage),X₁ is the total number of passengers arriving at the airport (measured in thousands),X₂ is an average of local hotel room rates,X₃ is the consumer confidence index,and X₄ is a dummy variable = 1 during the months of June,July,and August.He looks at the data from the past 36 months and obtains ln = 4.2 + 1.23lnx₁ - 2.2lnx₂ + 0.34ln x₃ + 2.3x₄ and R² = 0.63. -Interpret the estimate b₁.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: A professor investigated some of the factors that affect an individual student's final grade in his course.He proposed the multiple regression model Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε ,where Y is the final mark (out of 100),X₁ is the number of lectures skipped,X₂ is the number of late assignments,and X₃ is the mid-term test mark (out of 100).The professor recorded the data for 50 randomly selected students.The computer output is shown below. The regression equation is = 41.6 - 3.18x₁ - 1.17x₂ + 0.63x₃. S = 13.74 R-Sq = 30.0% ANALYSIS OF VARIANCE -Interpret the coefficient b₁.

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?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Suppose you are interested in determining the factors that influence the time required to prepare a tax return.You develop the model Y = β₀ + β₁X₁ + β₂X₂ + β₃ + β₄X₃ + β₅X₄ + β₆(X₁ ∙ X₄) + ε,where Y is the amount of time in minutes,X₁ is the income of the individual (in thousands of dollars),X₂ is the age of the individual,X₃ is the number of people living in the household,and X₄ is a dummy variable that takes the value 1 if the individual owns his or her own home.After interviewing 40 accountants,you obtained = 17.2 + 0.38x₁ + 1.04x₂ - 0.04 + 2.15x₃ + 0.11x₄ - 0.22(x₁ ∙ x₄), = 5.3, = 0.13, = 0.33, = 1.51, = 4.7, = 0.05,and = 0.07. -What are your alternative hypotheses regarding the individual coefficients in this model? Explain.

When you use the exponential model transformation,you need to take the natural log of all independent variables,including dummy variables.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: As director of the local tourist board,you are interested in determining the factors that influence the hotel occupancy rate in your city each month.Hotel occupancy can be measured as the percentage of available hotel rooms that are occupied by paying customers.You develop the following model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + ε,where Y is the hotel occupancy rate,X₁ is the total number of passengers arriving at the airport,X₂ is a price index of local hotel room rates,X₃ is the consumer confidence index,and X₄ is a dummy variable = 1 during the months of June,July,and August.You look at data from the past 36 months and obtain the following results: = 67.1 + 0.02x₁ - 0.055x₂ + 0.08x₃ + 12.3x₄,R² = 0.67, = 58.3, = 0.008, = 0.01, = 0.06, = 4.7,and SSE = 576. -Interpret the estimated regression coefficient b₄.

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