Exam 18: Model Building

In regression analysis,a nominal independent variable such as color,with three different categories such as red,white,and blue,is best represented by three indicator variables to represent the three colors.

It is not possible to incorporate nominal variables into a regression model.

The model y = $\beta$ ₀ + $\beta$ ₁x + $\beta$ ₂x² + $\varepsilon$ is referred to as a simple linear regression model.

Because indicator variables represent different groups,t-tests on the indicator variables allow us to draw inferences about the ____________________ in y between the groups.

In regression analysis,indicator variables are also called dependent variables.

The last category represented by I₁ = I₂ = .....I_{m - 1} = 0 is called the omitted category.

For the following regression equation $\tilde { y } = 20 + 8 x _ { 1 } + 5 x _ { 2 } + 3 x _ { 1 } x _ { 2 }$ ,which combination of x₁ and x₂,respectively,results in the largest average value of y?

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 price of a barrel of oil and the price of silver are linearly related?

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 odds ratio that an obese person who smokes 15 or more cigarettes per day suffers a heart attack is 9,then the probability that the person will suffer a heart attack is 0.81.

The model y = $\beta$ ₀ + $\beta$ ₁x + $\beta$ ₂x² +.........+ $\beta$ _px^p + $\varepsilon$ is referred to as a polynomial model with:

For the following regression equation $\tilde { y } = 15 + 6 x _ { 1 } + 5 x _ { 2 } + 4 x _ { 1 } x _ { 2 }$ ,a unit increase in x₁ increases the value of y on average by:

{Computer Training Narrative} Develop an estimated regression equation of the form $\tilde { y } = b _ { 0 } + b _ { 1 } x + b _ { 2 } x ^ { 2 }$ .

Regression analysis allows the statistics practitioner to use mathematical models to realistically describe relationships between the dependent variable and independent variables.

Which of the following statements about dummy variables is false?

Suppose that the sample regression equation of a second-order model is given by $\tilde { y } = 2.50 + 0.15 x + 0.45 x ^ { 2 }$ .Then,the value 2.50 is the:

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 ____________________ variable of a regression model is the variable that you wish to analyze or predict.

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