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Statistics For Business And Economics

James T. McClave, P. George Benson, Terry Sincich

$Math Studyset Vaia Explanations$ Math

13th Edition · 888 pages

ISBN: 9780134506593

Chapter 12: 86P (page 765)

Write a model that relates E(y) to two independent variables—one quantitative and one qualitative at four levels. Construct a model that allows the associated response curves to be second-order but does not allow for interaction between the two independent variables.

Short Answer

Expert verified

A second-order model with one quantitative variable and one qualitative variable with 4 levels can be written as $E (y) = β_{0} + β_{1} x_{1} + β_{2} {x_{1}}^{2} + β_{3} x_{2} + β_{4} x_{3} + β_{5} x_{4}$ .

Step by step solution

01

Variable conditions

There are two independent variables: one quantitative variable (say x₁) with a model in second-order and one qualitative variable with 4 levels (for k levels, (k-1) no of variables will be introduced in the model; namely x_2,x₃, and x₄). There are no interactions observed between the two independent variables.

02

Model for E(y)

A second-order model with one quantitative variable and one qualitative variable with 4 levels can be written as $E (y) = β_{0} + β_{1} x_{1} + β_{2} {x_{1}}^{2} + β_{3} x_{2} + β_{4} x_{3} + β_{5} x_{4}$

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Most popular questions from this chapter

Consider a multiple regression model for a response y, with one quantitative independent variable x₁ and one qualitative variable at three levels.

a. Write a first-order model that relates the mean response E(y) to the quantitative independent variable.

b. Add the main effect terms for the qualitative independent variable to the model of part a. Specify the coding scheme you use.

c. Add terms to the model of part b to allow for interaction between the quantitative and qualitative independent variables.

d. Under what circumstances will the response lines of the model in part c be parallel?

e. Under what circumstances will the model in part c have only one response line?

Catalytic converters in cars. A quadratic model was applied to motor vehicle toxic emissions data collected in Mexico City (Environmental Science & Engineering, Sept. 1, 2000). The following equation was used to predict the percentage (y) of motor vehicles without catalytic converters in the Mexico City fleet for a given year (x): ${\hat{β}}_{2}$

a. Explain why the value ${\hat{β}}_{0} = 325790$ has no practical interpretation.

b. Explain why the value ${\hat{β}}_{1} = - 321.67$ should not be Interpreted as a slope.

c. Examine the value of ${\hat{β}}_{2}$ to determine the nature of the curvature (upward or downward) in the sample data.

d. The researchers used the model to estimate “that just after the year 2021 the fleet of cars with catalytic converters will completely disappear.” Comment on the danger of using the model to predict y in the year 2021. (Note: The model was fit to data collected between 1984 and 1999.)

Question: Personality traits and job performance. Refer to the Journal of Applied Psychology (January 2011) study of the relationship between task performance and conscientiousness, Exercise 12.94 (p. 766). Recall that y = task performance score (measured on a 30-point scale) was modeled as a function of x₁ = conscientiousness score (measured on a scale of -3 to +3) and x₂ = {1 if highly complex job, 0 if not} using the complete model

$E (y) = β_{0} + β_{1} x_{1} + β_{2} {(x_{1})}^{2} + β_{3} x_{2} + β_{4} x_{1} x_{2} + β_{5} {(x_{1})}^{2} x_{2}$

a. Specify the null hypothesis for testing the overall adequacy of the model.

b. Specify the null hypothesis for testing whether task performance score (y) and conscientiousness score (x₁) are curvilinearly related.

c. Specify the null hypothesis for testing whether the curvilinear relationship between task performance score (y) and conscientiousness score (x₁) depends on job complexity (x₂).

Explain how each of the tests, parts a–c, should be conducted (i.e., give the forms of the test statistic and the reduced model).

Suppose you fit the second-order model $y = β_{0} + β_{1} x + β_{2} x^{2} + ε$ to n = 25 data points. Your estimate of $β_{2}$ is ${\hat{β}}_{2}$ = 0.47, and the estimated standard error of the estimate is 0.15.

Test $H_{0} : β_{2} = 0$ against $H_{a} : β_{2} \neq 0$ . Use $α = 0.05$ .
Suppose you want to determine only whether the quadratic curve opens upward; that is, as x increases, the slope of the curve increases. Give the test statistic and the rejection region for the test for $α = 0.05$ . Do the data support the theory that the slope of the curve increases as x increases? Explain.

Going for it on fourth down in the NFL. Refer to the Chance (Winter 2009) study of fourth-down decisions by coaches in the National Football League (NFL), Exercise 11.69 (p. 679). Recall that statisticians at California State University, Northridge, fit a straight-line model for predicting the number of points scored (y) by a team that has a first-down with a given number of yards (x) from the opposing goal line. A second model fit to data collected on five NFL teams from a recent season was the quadratic regression model, $E (y) = β_{0} + β_{1} x + β_{2} x^{2}$ .The regression yielded the following results: $y = 6.13 + 0.141 x - 0.0009 x^{2}$ , $R^{2} = 0.226$ .

a) If possible, give a practical interpretation of each of the b estimates in the model.

b) Give a practical interpretation of the coefficient of determination, $R^{2}$ .

c) In Exercise 11.63, the coefficient of correlation for the straight-line model was reported as $R^{2} = 0.18$ . Does this statistic alone indicate that the quadratic model is a better fit than the straight-line model? Explain.

d) What test of hypothesis would you conduct to determine if the quadratic model is a better fit than the straight-line model?

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