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Chapter 12: Q139SE (page 807)

Write a model relating E(y) to one qualitative independent variable that is at four levels. Define all the terms in your model.

Short Answer

Expert verified

Mathematically, the model in one qualitative independent variable with 4 levels can be written as E(y)=β0+β1x1+β2x2+β3x3

Step by step solution

01

A qualitative model

A model relating E(y) to one qualitative independent variable with 4 levels can be written by introducing (k-1) variables in the model to define the 4 different levels where 3 variables define 3 levels when the value of a particular variable is 1 and one level is defined as base level when all variables are 0.

02

4 levels

Mathematically, the model can be written asE(y)=β0+β1x1+β2x2+β3x3

Where x1= 1, if level 1 is present; otherwise 0

X2= 1, if level 2 is present; otherwise 0

X3 = 1, if level 3 is present; otherwise 0

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Most popular questions from this chapter

Suppose you fit the model y =β0+β1x1+β1x22+β3x2+β4x1x2+εto n = 25 data points with the following results:

β^0=1.26,β^1= -2.43,β^2=0.05,β^3=0.62,β^4=1.81sβ^1=1.21,sβ^2=0.16,sβ^3=0.26, sβ^4=1.49SSE=0.41 and R2=0.83

  1. Is there sufficient evidence to conclude that at least one of the parameters b1, b2, b3, or b4 is nonzero? Test using a = .05.

  2. Test H0: β1 = 0 against Ha: β1 < 0. Use α = .05.

  3. Test H0: β2 = 0 against Ha: β2 > 0. Use α = .05.

  4. Test H0: β3 = 0 against Ha: β3 ≠ 0. Use α = .05.

Personality traits and job performance. When attempting to predict job performance using personality traits, researchers typically assume that the relationship is linear. A study published in the Journal of Applied Psychology (Jan. 2011) investigated a curvilinear relationship between job task performance and a specific personality trait—conscientiousness. Using data collected for 602 employees of a large public organization, task performance was measured on a 30-point scale (where higher scores indicate better performance) and conscientiousness was measured on a scale of -3 to +3 (where higher scores indicate a higher level of conscientiousness).

a. The coefficient of correlation relating task performance score to conscientiousness score was reported as r = 0.18. Explain why the researchers should not use this statistic to investigate the curvilinear relationship between task performance and conscientiousness.

b. Give the equation of a curvilinear (quadratic) model relating task performance score (y) to conscientiousness score (x).

c. The researchers theorized that task performance increases as level of conscientiousness increases, but at a decreasing rate. Draw a sketch of this relationship.

d. If the theory in part c is supported, what is the expected sign ofβ2in the model, part b?

e. The researchers reportedβ^2=0.32with an associated p-value of less than 0.05. Use this information to test the researchers’ theory atα=0.05

Question: Novelty of a vacation destination. Many tourists choose a vacation destination based on the newness or uniqueness (i.e., the novelty) of the itinerary. The relationship between novelty and vacationing golfers’ demographics was investigated in the Annals of Tourism Research (Vol. 29, 2002). Data were obtained from a mail survey of 393 golf vacationers to a large coastal resort in the south-eastern United States. Several measures of novelty level (on a numerical scale) were obtained for each vacationer, including “change from routine,” “thrill,” “boredom-alleviation,” and “surprise.” The researcher employed four independent variables in a regression model to predict each of the novelty measures. The independent variables were x1 = number of rounds of golf per year, x2 = total number of golf vacations taken, x3 = number of years played golf, and x4 = average golf score.

  1. Give the hypothesized equation of a first-order model for y = change from routine.
  1. A test of H0: β3 = 0 versus Ha: β3< 0 yielded a p-value of .005. Interpret this result if α = .01.
  1. The estimate of β3 was found to be negative. Based on this result (and the result of part b), the researcher concluded that “those who have played golf for more years are less apt to seek change from their normal routine in their golf vacations.” Do you agree with this statement? Explain.
  1. The regression results for three dependent novelty measures, based on data collected for n = 393 golf vacationers, are summarized in the table below. Give the null hypothesis for testing the overall adequacy of the first-order regression model.
  1. Give the rejection region for the test, part d, for α = .01.
  1. Use the test statistics reported in the table and the rejection region from part e to conduct the test for each of the dependent measures of novelty.
  1. Verify that the p-values reported in the table support your conclusions in part f.
  1. Interpret the values of R2 reported in the table.

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): β^2

a. Explain why the valueβ^0=325790has no practical interpretation.

b. Explain why the valueβ^1=-321.67should not be Interpreted as a slope.

c. Examine the value ofβ^2to 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.)
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