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 -{Senior Medical Students Narrative} Interpret the coefficient b₃.

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 biology major student who has a 10.95 G.P.A.in their first three years (range from 0 to 12).

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

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 dependent variable must be ____________________.

Another term for a first-order polynomial model is a regression ____________________.

Silver Prices An economist is in the process of developing a model to predict the price of silver.She believes that the two most important variables are the price of a barrel of oil (x₁)and the interest rate (x₂).She proposes the first-order model with interaction: y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₁x₃ + $\varepsilon$ .A random sample of 20 daily observations was taken.The computer output is shown below. THE REGRESSION EQUATION IS y = 115.6 + 22.3x₁ + 14.7x₂ - 1.36x₁x₂ Predictar Coef StDev T Constant 115.6 78.1 1.480 22.3 7.1 3.141 14.7 6.3 2.333 -1.36 .52 -2.615 $S= 20.9 \quad R - S q = 55.4 \%$ ANALYSIS OF VARIANCE Source of Variation Regressian 3 8661 2887.0 6.626 Errorr 16 6971 435.7 Total 19 15632 -{Silver Prices Narrative} Interpret the coefficient b₁.

In the first-order model $\tilde { y } = 50 + 25 x _ { 1 } - 10 x _ { 2 } - 6 x _ { 1 } x _ { 2 }$ ,a unit increase in x₂,while holding x₁ constant at a value of 3,decreases the value of y on average by 3 units.

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 building a regression model,it is best to use the ____________________ number of independent variables that produce a satisfactory 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 -The stepwise regression procedure begins by computing the simple regression model for each independent variable.

The independent variable x in a polynomial model is called the ____________________ variable.

For the following regression equation $\tilde { y } = 10 + 3 x _ { 1 } + 4 x _ { 2 }$ ,a unit increase in x₂ increases the value of y on average by:

Motorcycle Fatalities A traffic consultant has analyzed the factors that affect the number of motorcycle fatalities.She has come to the conclusion that two important variables are the number of motorcycle and the number of cars.She proposed the model $\tilde { y } = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \varepsilon$ (the second-order model with interaction),where y = number of annual fatalities per county,x₁ = number of motorcycles registered in the county (in 10,000),and x₂ = number of cars registered in the county (in 1000).The computer output (based on a random sample of 35 counties)is shown below: THE REGRESSION EQUATION IS $y = 69.7 + 11.3 x _ { 1 } + 7.61 x _ { 2 } - 1.15 x _ { 1 } ^ { 2 } - 0.51 x _ { 2 } ^ { 2 } - 0.13 x _ { 1 } x _ { 2 }$ Predictor Coef StDev T Constant 69.7 41.3 1.688 11.3 5.1 2.216 7.61 2.55 2.984 -1.15 .64 -1.797 -.51 .20 -2.55 -.13 .10 -1.30 $S= 15.2 \quad \mathrm { R } - \mathrm { Sq } = 47.2 \%$ ANALYSIS OF VARIANCE Source of Variation Repressian 5 5959 1191.800 5.181 Error 29 6671 230.034 Total 34 12630 -{Motorcycle Fatalities Narrative} Test at the 1% significance level to determine if the x₂ term should be retained in the 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 -If the probability of an event is .20,then the odds ratio in favor of the event occurring is expressed as

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} Do these results allow us to conclude at the 1% significance level that the model is useful in predicting the income of physicians?

For the following regression equation $\tilde { y } = 50 + 10 x _ { 1 } - 4 x _ { 2 } - 6 x _ { 1 } x _ { 2 }$ ,a unit increase in x₂,while holding x₁ constant at a value of 3,decreases the value of y on average by:

If a quadratic relationship exists between y and each of x₁ and x₂,you use a(n)____________________-order polynomial model.

Motorcycle Fatalities A traffic consultant has analyzed the factors that affect the number of motorcycle fatalities.She has come to the conclusion that two important variables are the number of motorcycle and the number of cars.She proposed the model $\tilde { y } = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \varepsilon$ (the second-order model with interaction),where y = number of annual fatalities per county,x₁ = number of motorcycles registered in the county (in 10,000),and x₂ = number of cars registered in the county (in 1000).The computer output (based on a random sample of 35 counties)is shown below: THE REGRESSION EQUATION IS $y = 69.7 + 11.3 x _ { 1 } + 7.61 x _ { 2 } - 1.15 x _ { 1 } ^ { 2 } - 0.51 x _ { 2 } ^ { 2 } - 0.13 x _ { 1 } x _ { 2 }$ Predictor Coef StDev T Constant 69.7 41.3 1.688 11.3 5.1 2.216 7.61 2.55 2.984 -1.15 .64 -1.797 -.51 .20 -2.55 -.13 .10 -1.30 $S= 15.2 \quad \mathrm { R } - \mathrm { Sq } = 47.2 \%$ ANALYSIS OF VARIANCE Source of Variation Repressian 5 5959 1191.800 5.181 Error 29 6671 230.034 Total 34 12630 -{Motorcycle Fatalities Narrative} Test at the 1% significance level to determine if the x₁ term should be retained in the model.

An indicator variable (also called a dummy variable)is a variable that can assume either one of two values (usually 0 and 1),where one value represents the existence of a certain condition,and the other value indicates that the condition does not hold.

An indicator variable is a nominal variable that can assume _______________ possible values.

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 a multiple regression analysis,which procedure permits variables to enter and leave the model at different stages of its development?