Exam 12: Multiple Regression

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:

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. -Set up the analysis of variance table for testing H₀ : β₁ = β₂ = 0 against H₁ : At least one β_j ≠ 0,(j = 1,2).

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS: = 55.8 + 1.79x₁ - 0.021x₂ - 0.016x₃ S = 9.47 R-Sq = 22.5% ANALYSIS OF VARIANCE -Interpret the coefficient b₃.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS: = 55.8 + 1.79x₁ - 0.021x₂ - 0.016x₃ S = 9.47 R-Sq = 22.5% ANALYSIS OF VARIANCE -Is there sufficient evidence at the 5% significance level to infer that the number of points that the individual's blood pressure exceeded the recommended value and the age at death show a negative linear relationship?

Determine the coefficient of multiple correlation.

The confidence intervals and the hypothesis tests in multiple regression are not seriously affected by departures from normality in the distribution of the error terms.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: A regression analysis has produced the following partial analysis of variance table: Analysis of Variance -Compute SST.

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. -Test H₀ : β₄ = 0 against H₁ : β₄ ≠ 0 and interpret your result.

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. -Test the hypothesis H₀ : β₁ = 0 and interpret your results.

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. -What are your hypotheses regarding this model? Explain.Are there any other variables you would have liked to include?

How should the loan officer interpret the coefficient on x₂?

How would we interpret the coefficient of determination R²?

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

In multiple regression analysis,to test H₀ : β₁ = β₂ = ∙ ∙ ∙ ∙ ∙ ∙ = β_k = 0,an F statistic with K and n - K - 1 degrees of freedom is used.

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

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. -Test the hypothesis H₀ : β₃ = 0 and interpret your result.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: An estimated linear model is given by = 12 - 3x₁ - 4x₂ + 7x₃. -When x₂ decreases by 2,what is the change in ?

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?

In multiple regression analysis,the mean square regression (MSR)is a measure of the explained variability adjusted for the number of independent variables.

As the sample size increases,we would expect a greater difference between R² and ².