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 -The two largest values in a correlation matrix are the .89 correlation between y and x₃,and the .83 correlation between y and x₇.During a stepwise regression analysis x₃ is the first independent variable brought into the equation.Will x₇ necessarily be next? If not,why not?

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 the model is useful in predicting the fourth-year medical course final grade?

In the first-order regression model $\tilde { y } = 12 + 6 x _ { 1 } + 8 x _ { 2 } + 4 x _ { 1 } x _ { 2 }$ ,a unit increase in x₁ increases the value of y on average by 6 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 -One of the requirements of regression analysis is that the dependent variable must be:

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 on average biology majors outperform those whose majors are not medical or biology?

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 interaction term should be retained in the model.

For the following regression equation $\tilde { y } = 75 + 20 x _ { 1 } - 15 x _ { 2 } + 5 x _ { 1 } x _ { 2 }$ ,a unit increase in x₂,while holding x₁ constant at 1,changes the value of y on average by:

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} Is there sufficient evidence at the 1% significance level to conclude that the interaction term should be retained?

In explaining starting salaries for graduates of psychology programs,which of the following independent variables would not best be represented with dummy variables?

Suppose that the sample regression equation of a model is $\tilde { y } = 10 + 4 x _ { 1 } + 3 x _ { 2 } - x _ { 1 } x _ { 2 }$ .If we examine the relationship between x₁ and y for three different values of x₂,we observe that the:

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

Which of the following is not an advantage of multiple regression as compared with analysis of variance?

A second-order polynomial model is shaped like a(n)____________________.

The model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\varepsilon$ is referred to as a first-order model with two predictor variables with no interaction.

In explaining the income earned by college graduates,which of the following independent variables is best represented by a dummy variable?

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} Is there sufficient evidence at the 1% significance level to conclude that the interest rate and the price of silver are linearly related?

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} What does the coefficient of $x _ { 2 } ^ { 2 }$ tell you about the model?

It is possible to include ____________________ variables in a regression model.This is accomplished through the use of indicator (or dummy)variables.

In explaining the amount of money spent on gifts for a child's birthday each year,the independent variable,age of child,is best represented by a dummy variable.

An indicator variable can assume either one of only two values (usually 0 and 1),where _______________ represents the existence of a certain condition and _______________ indicates that the condition does not hold.