Exam 20: Model Building

A regression analysis was performed to study the relationship between a dependent variable and four independent variables. The following information was obtained: r² = 0.95, SSR = 9800, n = 50. ANOVA Source of Variation df SS MS F Regression 4 9800 2450 50.114 Error 45 2200 48.889 Total 49 12,000 Test the overall validity of the model at the 5% significance level.

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

An avid football fan was in the process of examining the factors that determine the success or failure of football teams. He noticed that teams with many rookies and teams with many veterans seem to do quite poorly. To further analyse his beliefs, he took a random sample of 20 teams and proposed a second-order model with one independent variable. The selected model is: $y = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } + \varepsilon$ . where y = winning team's percentage. x = average years of professional experience. The computer output is shown below: THE REGRESSION EQUATION IS: $y =$ $32.6 + 5.96 x - 0.48 x ^ { 2 }$ Predictor Coef S2Dev T Constant 32.6 19.3 1.689 x 5.96 2.41 2.473 -0.48 0.22 -2.182 S = 16.1 R-Sq = 43.9%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 2 3452 1726 6.663 Error 17 4404 259.059 Total 19 7856 Do these results allow us to conclude at the 5% significance level that the model is useful in predicting the team's winning percentage?

In explaining starting salaries for graduates of computer science programs, which of the following independent variables would not be adequately represented with a dummy variable?

Consider the following data for two variables, x and y, where x is the age of a particular make of car and y is the selling price, in thousands of dollars. x 7 10 3 5 3 10 4 14 5 8 y 35.0 28.5 45.0 45.0 55.0 25.0 37.5 27.5 30.0 27.5 Use Excel to test whether the population slope is positive, at the 1% level of significance.

In a stepwise regression procedure, if two independent variables are highly correlated, then:

A professor of accounting wanted to develop a multiple regression model to predict the students' grades in her fourth-year accounting course. She decides that the two most important factors are the student's grade point average (GPA) in the first three years and the student's major. She proposes the model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . where y = fourth-year accounting course mark (out of 100). $x _ { 1 }$ = GPA in first three years (range 0 to 12). $x _ { 2 }$ = 1 if student's major is accounting. = 0 if not. $X _ { 3 }$ = 1 if student's major is finance. = 0 if not. The computer output is shown below. THE REGRESSION EQUATION IS $y =$ $9.14 + 6.73 x _ { 1 } + 10.42 x _ { 2 } + 5.16 x _ { 3 }$ . Predictor Coef SDev 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 R-Sq = 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 Do these results allow us to conclude at the 1% significance level that on average finance majors outperform those whose majors are not accounting or finance?

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

A first-order model was used in a regression analysis involving 25 observations to study the relationship between a dependent variable y and three independent variables, $x _ { 1 }$ , $x _ { 2 }$ and $X _ { 3 }$ . The analysis showed that the mean squares for regression is 160 and the sum of squares for error is 1050. In addition, the following is a partial computer printout. Predictor Coef StDev Constant 25 4 18 6 -12 4.8 6 5 Is there sufficient evidence at the 5% significance level to indicate that $x _ { 2 }$ is negatively linearly related to y?

A traffic consultant has analysed the factors that affect the number of traffic fatalities. She has come to the conclusion that two important variables are the number of cars and the number of tractor-trailer trucks. She proposed the second-order model with interaction: $y =$ $\beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \varepsilon$ . Where: y = number of annual fatalities per shire. $x _ { 1 }$ = number of cars registered in the shire (in units of 10 000). $x _ { 2 }$ = number of trucks registered in the shire (in units of 1000). The computer output (based on a random sample of 35 shires) 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 SDev T Constant 69.7 41.3 1.688 11.3 5.1 2.216 7.61 2.55 2.984 -1.15 0.64 -1.797 -0.51 0.20 -2.55 -0.13 0.10 -1.30 S = 15.2 R-Sq = 47.2%. ANALYSIS OF VARIANCE Source of df SS MS F Variation Regression 5 5959 1191.800 5.181 Error 29 6671 230.034 Total 34 12630 Test at the 1% significance level to determine whether the interaction term should be retained in the model.

In explaining the income earned by university graduates, which of the following independent variables is best represented by an indicator variable in a regression model?

A professor of accounting wanted to develop a multiple regression model to predict the students' grades in her fourth-year accounting course. She decides that the two most important factors are the student's grade point average (GPA) in the first three years and the student's major. She proposes the model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . where y = fourth-year accounting course mark (out of 100). $x _ { 1 }$ = GPA in first three years (range 0 to 12). $x _ { 2 }$ = 1 if student's major is accounting. = 0 if not. $X _ { 3 }$ = 1 if student's major is finance. = 0 if not. The computer output is shown below. THE REGRESSION EQUATION IS $y =$ $9.14 + 6.73 x _ { 1 } + 10.42 x _ { 2 } + 5.16 x _ { 3 }$ . Predictor Coef SDev 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 R-Sq = 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 Interpret the coefficient $b _ { 3 }$ .