Question

Agreeableness, gender, and wages. 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 (February 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, E1y2, as a function of both agreeableness score 1x12 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.

a) Consider the model,$\mathit{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$. 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 b1 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,$\mathit{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$. 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 b coefficients. How do they compare with 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 a = .05. Is the researchers’ theory supported?

Short answer

a) If income decreases for either gender as agreeable score increases the sign of β₁should be negative as a negative relation between dependent and independent variable will be expressed by a negative sign of coefficient of independent variable.

b) The researchers’ theory that the rate of decrease of income with agreeableness score would be steeper for males than females can be tested by testing the significance of β₂. If β₂is more than 1 then it indicates that rate of decrease of income with agreeableness score would be steeper for males than females.

c) In interaction model, the rate of decrease of income with agreeableness score would be steeper for males than for females will be represented by a negative sign of β₁and the sign of β₃ will be negative given that due to the interaction between x₁(agreeableness score) and x₂(gender) results in decrease in mean level of income.

d) From the excel output it can be said that the sign of β₁is negative as expected but the sign of β₃ is positive indicating positive relation between the interaction and mean level of incomes.

e) To test whether the rate of decrease of income with agreeableness score is steeper for males than for females the null hypothesis would be; H₀: β₃= 0 while the alternate hypothesis would be H₁: β₃> 0.

f) At 95% confidence level,β₃= 0 therefore the rate of decrease of income with agreeable score is not steeper for males than females.

Step-by-step solution

Sign of β1

If income decreases for either gender as agreeable score increases the sign of β₁should be negative as a negative relation between dependent and independent variable will be expressed by a negative sign of coefficient of independent variable.

Testing of β2

The researchers’ theory that the rate of decrease of income with agreeableness score would be steeper for males than females can be tested by testing the significance of β₂. If β₂is more than 1 then it indicates that rate of decrease of income with agreeableness score would be steeper for males than females.

Sign of β1 and β3

In interaction model, the rate of decrease of income with agreeableness score would be steeper for males than for females will be represented by a negative sign of β₁and the sign of β₃ will be negative given that due to the interaction between x₁(agreeableness score) and x₂(gender) results in decrease in mean level of income.

Model

The excel output is presented below

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.974883
R Square	0.950396
Adjusted R Square	0.925594
Standard Error	3773.926
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	3	1.64E+09	5.46E+08	38.31952	0.000262
Residual	6	85455111	14242519
Total	9	1.72E+09
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	48520.96	14151.72	3.428627	0.013995	13892.95	83148.97	13892.95	83148.97
Agree score(x1)	-8667.38	4363.577	-1.9863	0.094198	-19344.7	2009.907	-19344.7	2009.907
Gender(x2)	18900.74	22516.8	0.839406	0.433407	-36195.9	73997.37	-36195.9	73997.37
X1*x2	1926.729	6927.41	0.278131	0.790248	-15024	18877.49	-15024	18877.49

From the excel output it can be said that the sign of β₁is negative as expected but the sign of β₃ is positive indicating positive relation between the interaction and mean level of incomes.

Significance of β3

To test whether the rate of decrease of income with agreeableness score is steeper for males than for females the null hypothesis would be; H₀: β₃= 0 while the alternate hypothesis would be H₁: β₃> 0.

Interpretation of β3

$H_0:{\beta }_3=0$while $H_a:{\beta }_3>0$

The p-value of β₃ is 0.790248 from the excel output while α = 0.05.

$H_0$is rejected if $p$value <α . For α =0.05, since 0.790248>0.05

Not sufficient evidence to reject $H_0$at 95% confidence interval

Therefore,${\beta }_3=0$