Exam 12: Multiple Regression

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 ?

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

In reference to the multiple regression model = 40 + 15x₁ - 10x_{2 +} 5x₃,if x₂ were to increase by five units,holding x₁ and x₃ constant,then the value of y would decrease on average by 50 units.

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. -Test the hypothesis H₀ : β₁ = β₂ = 0 against H₁ : At least one β_j ≠ 0,(j = 1,2)at the 1% significance level.

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. -Interpret the estimate b₂.

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 enough evidence at the 10% significance level to infer that the model is useful in predicting how long an individual will live?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Consider the following regression model: Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε.Suppose that you have estimated the coefficients for the model,and obtained the partial ANOVA table shown below. Analysis of Variance -Test the null hypothesis H₀ : β₁ = β₂ = β₃ = 0 against H₁ : At least one β_j ≠ 0,j = 1,2,3 at the 1% significance level.

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. -Is it possible to provide a meaningful interpretation of the estimate b₀?

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

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 -Does the data provide enough evidence to conclude that,at the 5% significance level,the final mark and the number of late assignments are negatively linearly related?

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 = β₂ = β₃ = β₄ = 0.Interpret your results.

In multiple regression,if the confidence interval for a regression parameter β_i includes 0,we cannot reject the two-sided hypothesis that this coefficient is 0.

By transforming the variables,we can estimate a linear multiple regression model and use the results as a nonlinear model.

The coefficient of multiple correlation R measures the strength of the relationship between the dependent variable and the independent 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 b₂.

The least squares procedure for multiple regression computes the estimated coefficients so as to minimize the sum of the residuals.

Adding independent variables to the multiple regression model increases the value of the coefficient of determination.

Determine the adjusted coefficient of determination.

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:

The value of b₃ is: