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 -Indicator variables can assume as many values as there are categories of its corresponding nominal variable.

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} Estimate the annual income for a general practitioner with 15 years of experience.

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 stepwise regression,the independent variable with the largest F-statistic,or equally the smallest p-value,is chosen as the first entering variable.

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 x² term should be retained.

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 -What is stepwise regression,and when is it desirable to make use of this multiple regression technique?

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 -To gather the required observations for your potential regression models,a general rule is that there should be at least ____________________ observations for each independent variable used in the equation.

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} Estimate the annual income for a surgeon with 15 years of experience.

The model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₁x₂ + $\varepsilon$ is referred to as a second-order model with two predictor variables with interaction.

In a first-order polynomial model with no interaction,the effect of x₁ on y remains the same no matter what the value of x₂ is.The graph of this model produces straight lines that are ____________________ to each other.

When we plot x versus y,the graph of the model y = $\beta$ ₀ + $\beta$ ₁x + $\beta$ ₂x² + $\varepsilon$ is shaped like a:

For the following regression equation $\tilde { y } = 100 - 12 x _ { 1 } + 5 x _ { 2 } - 4 x _ { 1 } x _ { 2 }$ ,a unit increase in x₁,while holding x₂ constant at a value of 2,decreases the value of y on average by:

The model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\varepsilon$ is referred to as a:

The model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₁x₂ + $\varepsilon$ is referred to as a:

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 -Another name for an indicator variable is an interval variable.

{Computer Training Narrative} Determine if there is sufficient evidence at the 5% significance level to infer that the quadratic relationship between y,x,and x² in the previous question is significant.

In the first-order model $\hat { y } = 8 + 3 x _ { 1 } + 5 x _ { 2 }$ ,a unit increase in x₂,while holding x₁ constant,increases the value of y on average by 5 units.

{Computer Training Narrative} Determine if there is sufficient evidence at the 5% significance level to infer that the relationship between x and y is positive and significant.

The model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₁x₂ + $\varepsilon$ is a(n)____________________-order polynomial model with ____________________ predictor variables and ____________________.

{Computer Training Narrative} Estimate the value of y when x = 45 using the estimated linear regression equation in the previous question.

It is possible to include nominal variables in a regression model.This is accomplished through the use of ____________________ variables,also known as ____________________ variables.