Exam 18: Model Building

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -In multiple regression,which procedure permits variables to enter and leave the model at different stages of its development?

Hockey Teams An avid hockey fan was in the process of examining the factors that determine the success or failure of hockey teams.He noticed that teams with many rookies and teams with many veterans seem to do quite poorly.To further analyze his beliefs he took a random sample of 20 teams and proposed a second-order model with one independent variable,average years of professional experience.The selected model is y = $\beta$ ₀ + $\beta$ ₁x + $\beta$ ₂x² + $\varepsilon$ ,where y = winning team's percentage,and x = average years of professional experience.The computer output is shown below. THE REGRESSION EQUATION IS y = 32.6 + 5.96x - .48x² Predictar Coef StDev Constant 32.6 19.3 1.689 5.96 2.41 2.473 -.48 .22 -2.182 S = 16.1 $\quad$ R - S q = 43.9 % ANALYSIS OF VARIANCE Source of Variation Regressian 2 3452 1726 6.663 Error 17 4404 259.059 Total 19 7856 -{Hockey Teams Narrative} Test to determine at the 10% significance level if the linear term should be retained.

In the first-order model $\tilde { y } = 60 + 40 x _ { 1 } - 10 x _ { 2 } + 5 x _ { 1 } x _ { 2 }$ ,a unit increase in x₂,while holding x₁ constant at 1,changes the value of y on average by -5 units.

Suppose that the sample regression equation of a model is $\tilde { y } = 4.7 + 2.2 x _ { 1 } + 2.6 x _ { 2 } - x _ { 1 } x _ { 2 }$ .If we examine the relationship between y and x₂ for x₁ = 1,2,and 3,we observe that the three equations produced not only differ in the intercept term,but the coefficient of x₂ also varies.

A dummy variable is used as an independent variable in a regression model when:

{Computer Training Narrative} Determine the coefficient of determination quadratic model.What does this statistic tell you about this model?

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -{Senior Medical Students Narrative} Interpret the coefficient b₂.

When a dummy variable is included in a multiple regression model,the interpretation of the estimated slope coefficient does not make any sense anymore.

Suppose that the sample regression line of the first-order model is $\hat { y } = 8 + 2 x _ { 1 } + 3 x _ { 2 }$ .If we examine the relationship between y and x₁ for four different values of x₂,we observe that the:

Incomes of Physicians An economist is analyzing the incomes of physicians (general practitioners,surgeons,and psychiatrists).He realizes that an important factor is the number of years of experience.However,he wants to know if there are differences among the three professional groups.He takes a random sample of 125 physicians and estimates the multiple regression model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = annual income (in $1,000),x₁ = years of experience,x₂ = 1 if physician and 0 if not,and x₃ = 1 if surgeons and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 71.65 + 2.07x₁ + 10.16x₂ - 7.44x₃ Predictar Codef StDev T Constant 71.65 18.56 3.860 2.07 .81 2.556 10.16 3.16 3.215 -7.44 2.85 -2.611 $S = 42.6 \quad R - S q = 30.9 \%$ ANALYSIS OF VARIANCE -{Incomes of Physicians Narrative} Is there enough evidence at the 1% significant level to conclude that general practitioners earn more on average than psychiatrists?

Hockey Teams An avid hockey fan was in the process of examining the factors that determine the success or failure of hockey teams.He noticed that teams with many rookies and teams with many veterans seem to do quite poorly.To further analyze his beliefs he took a random sample of 20 teams and proposed a second-order model with one independent variable,average years of professional experience.The selected model is y = $\beta$ ₀ + $\beta$ ₁x + $\beta$ ₂x² + $\varepsilon$ ,where y = winning team's percentage,and x = average years of professional experience.The computer output is shown below. THE REGRESSION EQUATION IS y = 32.6 + 5.96x - .48x² Predictar Coef StDev Constant 32.6 19.3 1.689 5.96 2.41 2.473 -.48 .22 -2.182 S = 16.1 $\quad$ R - S q = 43.9 % ANALYSIS OF VARIANCE Source of Variation Regressian 2 3452 1726 6.663 Error 17 4404 259.059 Total 19 7856 -{Hockey Teams Narrative} What is the coefficient of determination? Explain what this statistic tells you about the model.

We interpret the coefficients in a multiple regression model by holding all variables in the model constant.

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -{Senior Medical Students Narrative} Predict the final grade (out of 100)in the fourth-year medical course for a medical student who has a 10.95 G.P.A.in their first three years (range from 0 to 12).

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -{Senior Medical Students Narrative} Do these results allow us to conclude at the 1% significance level that grade point average in first three years is linearly related to the fourth-year medical final grade?

Color of truck is a(n)____________________ variable.

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -When the dependent variable is nominal,a(n)____________________ regression model is used.

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -(Senior Medical Students Narrative)Predict the final grade (out of 100)in the fourth-year medical course for an anatomy major student who has a 10.95 G.P.A.in their first three years (range from 0 to 12).

In order to represent a nominal variable with m categories,we must create m - 1 indicator variables.

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -The stepwise regression procedure begins by computing the multiple regression model for all independent variables of interest.

Suppose that the sample regression line of the first-order model is $\hat { y } = 4 + 3 x _ { 1 } + 2 x _ { 2 }$ .If we examine the relationship between y and x₁ for three different values of x₂,we observe that the effect of x₁ on y remains the same no matter what the value of x₂.