Question

Data was collected from 40 employees to develop a regression model to predict the employee’s annual salary using their years with the company (Years), their starting salary in thousands (Starting), and their Gender (Male = 0, Female = 1). The level of significance is .01. The results from Excel regression analysis are shown below:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.718714957
R Square	0.516551189
Adjusted R Square	0.476263788
Standard Error	10615.63461
Observations	40
ANOVA
	Df	SS	MS	F	Significance F
Regression	3	4334682510	1444894170	12.82165585	7.48476E-06
Residual	36	4056901131	112691698.1
Total	39	8391583641
	Coefficients	Standard Error	t Stat	P-value
Intercept	27946.57894	4832.438706	5.783121245	1.35464E-06
Years	1665.251558	425.0829092	3.917474737	0.000383313
Starting	0.266374185	0.12610443	2.112330112	0.041661598
Gender	-3285.541043	5617.145392	-0.584912943	0.56225464

1. Is the overall model statistically significant? Why or why not?
2. Is the slope coefficient on starting salary statistically significant? You must justify your answer.
3. Find the predicted starting salary for a male employee with 7 years of experience who had a starting salary of $10,000.
4. Calculate the marginal effect of 3 additional work years on salary?
5. Do the results provide evidence of gender discrimination? Why or Why not?

Answer 1

Let the regression model being estimated be

a) Is the overall model statistically significant? Why or why not?

The hypotheses to test if the overall model statistically significant are

Using the following

The test statistic is F=12.82

The p-value is 0.0000 (rounded to 4 decimals)

Using the level of significance

we will reject the null hypothesis, if the p-value is less than the level of significance.

Here, the p-value is 0.0000 and it is less than 0.01. Hence we reject the null hypothesis.

ans: Reject H₀. The overall model statistically significant as the p-value is less than the level of significance.

b) Is the slope coefficient on starting salary statistically significant? You must justify your answer.

The slope coefficient of the starting salary is in the population regression model.

To test if the slope coefficient on starting salary statistically significant, we test the following hypotheses

Using the following

we get

the test statistic is t=2.112 and the p-value = 0.0417

Using the level of significance

we will reject the null hypothesis, if the p-value is less than the level of significance.

Here, the p-value is 0.0417 and it is not less than 0.01. Hence we do not reject the null hypothesis.

ans: Fail to Reject H₀. The slope coefficient on starting salary is not statistically significant as the p-value is not less than the level of significance.

c) Find the predicted starting salary for a male employee with 7 years of experience who had a starting salary of $10,000.

Using this

The estimated regression equation is

the predicted Annual salary for a male employee (Gender=0) with 7 years of experience (Years=7) who had a starting salary of $10,000 (Starting=10 (in 1000s)) is

ans: the predicted annual salary for a male employee with 7 years of experience who had a starting salary of $10,000 is $39,606.00

Note: the change from "the predicted starting salary for..." to "the predicted annual salary for..." as the regression model is to predict the employee’s annual salary using their years with the company (Years), their starting salary in thousands (Starting), and their Gender (Male = 0, Female = 1) as given in the question.

d) Calculate the marginal effect of 3 additional work years on salary?

The estimated slope coefficient of Years is 1665.2516. The positive value tells us that the Annual salary and the Years move in the same direction. It says that for each additional work year, the annual salary increases by $1,665.2516, when keeping other variables unchanged. Hence we can say that for 3 additional work years, the salary would increase by 3*1665.2516=4995.75

ans: the marginal effect of 3 additional work years on salary is $4995.75

e) Do the results provide evidence of gender discrimination? Why or Why not?

There would be evidence of gender discrimination, if the slope coefficient of Gender (which is ) is not equal to 0.

The hypotheses to test this are

Using this

the test statistic is t=-0.585 and the p-value is 0.5623

Using the level of significance

we will reject the null hypothesis, if the p-value is less than the level of significance.

Here, the p-value is 0.5623 and it is not less than 0.01. Hence we do not reject the null hypothesis.

ans: Fail to Reject H₀.  There is no sufficient evidence to conclude that the slope coefficient of Gender is not equal to zero. The results do not provide evidence of gender discrimination.