Exam 18: Model Building

In general,to represent a nominal variable with m categories,we must create _______________ indicator variables.The last category represented by I₁ = I₂ = .....I_{m - 1} = 0 is called the _______________ category.

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} Do these results allow us at the 5% significance level to conclude that the model is useful in predicting the price of silver?

A first-order polynomial model with one predictor variable is the familiar simple linear regression 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} Do these results allow us to conclude at the 1% significance level that on average medical majors outperform those whose majors are not medical or biology?

Suppose that the sample regression equation of a model is $\tilde { y } = 4 + 1.5 x _ { 1 } + 2 x _ { 2 } - x _ { 1 } x _ { 2 }$ .If we examine the relationship between x₁ and y for four different values of x₂,we observe that the four equations produced differ only in the intercept term.

In explaining students' test scores,which of the following independent variables would not best be represented with indicator variables?

In a first-order model with two predictors x₁ and x₂,an interaction term may be used when the relationship between the dependent variable y and the predictor variables is linear.

{Computer Training Narrative} Develop a scatter diagram for the data.Does the scatter diagram suggest an estimated regression equation of the form $\tilde { y } = b _ { 0 } + b _ { 1 } x + b _ { 2 } x ^ { 2 }$ ? Explain.

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,if two independent variables are highly correlated,both variables must enter the model simultaneously.

Computer Training Consider the following data for two variables,x and y.The independent variable x represents the amount of training time (in hours)for a salesperson starting in a new computer store to adjust fully,and the dependent variable y represents the weekly sales (in $1000s). 10 14 16 20 25 30 35 40 50 12 20 23 27 36 45 40 28 30 Use statistical software to answer the following question(s). -{Computer Training Narrative} Develop an estimated regression equation of the form $\hat { y } = b _ { 0 } + b _ { 1 } x$ .

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 -____________________ regression is an iterative procedure that adds and deletes one independent variable at a time to/from the regression 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} Is there enough evidence at the 5% significance level to conclude that the model is useful in predicting the number of fatalities?

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 -At each step of the stepwise regression procedure,the p-values of all variables are computed and composed to the F-to-remove.If a variable's F-statistic falls below this standard,it is removed from the equation.

In explaining the amount of money spent on children's shoes each month,which of the following independent variables is best represented with an indicator variable?

____________________ means that the effect of x₁ on y is influenced by the value of x₂,and vice versa.

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 $x _ { 1 } ^ { 2 }$ term should be retained in the model.

The graph of the model $\tilde { y } _ { i } = \beta _ { 0 } + \beta _ { 1 } x _ { i } + \beta _ { 2 } x _ { i } ^ { 2 }$ is shaped like a straight line going upwards.

Which of the following statements is false regarding the graph of the second-order polynomial model y = $\beta$ ₀ + $\beta$ ₁x + $\beta$ ₂x² + $\varepsilon$ ?

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 5% significance level to conclude that income and experience are linearly related?

In the first-order model $\tilde { y } = 75 - 12 x _ { 1 } + 5 x _ { 2 } - 3 x _ { 1 } x _ { 2 }$ ,a unit increase in x₁,while holding x₂ constant at a value of 2,decreases the value of y on average by 8 units.