Do agreeable individuals get paid less, on average, than those who are less agreeable on the job? And is this gap greater for males than for females? These questions were addressed in the Journal of Personality and Social Psychology (Feb. 2012). Several variables were measured for each in a sample of individuals enrolled in the National Survey of Midlife Development in the U.S. Three of these variables are: (1) level of agreeableness score (where higher scores indicate a greater level of agreeableness), (2) gender (male or female), and (3) annual income (dollars). 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). Data for a sample of 100 individuals (simulated, based on information provided in the study) are saved in the file. The first 10 observations are listed in the accompanying table.
Data for First 10 Individuals in Study

a. Consider the model, E(y) = β0 + β1x1 + β2x2. The researchers theorized that for either gender, income would decrease as agreeableness score increases. If this theory is true, what is the expected sign of β1 in the model?
b. The researchers also theorized that the rate of decrease of income with agreeableness score would be steeper for males than for females (i.e., the income gap between males and females would be greater the less agreeable the individuals are). Can this theory be tested using the model, part a? Explain.
c. Consider the interaction model, E(y) = β0 + β1x1 + β2x2 + β3x1x2. If the theory, part b, is true, give the expected sign of β1. The expected sign of β3.
d. Fit the model, part c, to the sample data. Check the signs of the estimated β coefficients. How do they compare to the expected values, part c?
e. Refer to the interaction model, part c. Give the null and alternative hypotheses for testing whether the rate of decrease of income with agreeableness score is steeper for males than for females.
f. Conduct the test, part e. Use α = .05. Is the researchers' theory supported?

Income Agree Score Gender 44,770 51,480 39.600 24.370 15,460 43,730 48,330 25,970 17120 20,140 3.0 3.3 3.3 3.6 3.8 3.2 3.5 3.2