Inequalities in the annual salary for employees with similar performance ratings, years of service, and certifications

The personnel director for a local manufacturing firm has received complaints from the employees in a certain shop regarding what they perceive to be inequities in the annual salary for employees who have similar performance ratings, years of service, and relevant certifications. The personnel director believes that an employee's pay in this particular shop should be positively correlated to their prior performance rating, years of service, and relevant certifications. The personnel director has collected the data shown in the following table pertaining to the employees within the shop:

Employee Current Annual Salary Average Performance Years of Service Number of Relevant
(Thousands) Rating for Past 3 Years Certifications
(5 point scale)

1 56.1 2.18 9 6
2 55.3 3.31 21 7
3 48.9 3.18 18 7
4 61.8 3.75 36 7
5 56.4 2.62 31 6
6 52.5 3.75 15 6
7 52.6 4.25 25 6
8 62.6 2.43 30 5
9 45.1 1.93 7 6
10 71.1 3.50 47 8
11 53.2 2.81 26 6
12 44.3 3.06 11 6
13 55.3 5.00 19 6
14 59.1 4.06 35 7
15 60.0 4.12 38 9
16 48.6 5.00 21 4
17 50.4 3.87 9 6
18 63.0 4.37 41 8
19 53.0 2.50 35 3
20 50.9 2.81 23 4
21 55.4 3.68 33 4
22 51.8 5.00 27 4
23 62.0 3.00 37 8
24 50.1 2.43 15 5

The personnel director is interested in creating a linear regression model that can be used to estimate the annual salary an employee might expect to receive based upon his or her past performance, years of service, and/or number of relevant certifications. The regression model will be used as a basis for determining whether or not there is any validity to the employees' complaints regarding salary inequities.

Perform each of the following seven regression analyses using a 95% confidence level.
- Annual salary vs. average performance rating for the past 3 years
- Annual salary vs. years of service
- Annual salary vs. number of relevant certifications
- Annual salary vs. average performance rating for the past 3 years and years of service
- Annual salary vs. average performance rating for the past 3 years and number of relevant certifications
- Annual salary vs. years of service and number of relevant certifications
- Annual salary vs. average performance rating for the past 3 years, years of service and number of relevant certifications

Use the results for the univariate regression analysis for annual salary vs. average performance rating for the past 3 years in order to answer questions 1 through 8.

1. What is the degree of correlation between the dependent variable and the independent variable?

2. Is the statistical significance of the model as a whole less than the desired statistical significance for the regression model? Explain the basis for your answer.

3. Is the statistical significance of the linear relationship between the dependent and independent variables less than the desired statistical significance for the regression model? Explain the basis for your answer.

4. What percentage of the observed variation between the actual values of the dependent variable and the mean value of the dependent variable in the sample data set is explained by the regression model?

5. What is the amount by which we will be off on average when predicting values for the dependent variable using the regression model?

6. What is the coefficient for the y-intercept for the regression model?

7. What is the coefficient for the independent variable for the regression model?

8. What is the point estimate for the predicted salary for an employee with an average performance rating of 3.9?