Refer to the Journal of Personality and Social Psychology (February 2012) study of on-the-job agreeableness and wages, Exercise 12.92. The researchers modeled mean income, E(y), as a function of both agreeableness score (x1) and a dummy variable for gender (x2 = 1 if male, 0 if female). Suppose the researchers theorize that for either gender, income will decrease at a decreasing rate as agreeableness score increases. Consequently, they want to fit a second-order model.

a. Consider the model, E(y) = β0 + β1x1 + β2(x1)2 + β3x2. If the researchers' belief is true, what is the expected sign of β2 in the model?

b. Draw a sketch of the model, part a, showing how gender impacts the income-agreeableness score relationship.

c. Write a complete second-order model for E(y) as a function of x1 and x2.

d. Draw a sketch of the model, part c, showing how gender impacts the income-agreeableness score relationship.

e. What null hypothesis would you test in order to compare the two models, parts a and c?

f. Fit the models to the sample data saved in the file and carry out the test, part e. What do you conclude? (Test using α = .10.)