The personnel director for a local manufacturing firm has received complaints from the employees in a certain

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Question:

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

Hints:

• Refer to the handouts posted on Blackboard pertaining to interpreting regression statistics in order to determine if a given regression model is acceptable. This same handout also provides guidance regarding how to select a preferred regression model from amongst multiple acceptable regression models, including models with differing numbers of independent variables.

• For the purposes of this homework assignment, for question 13 the minimum difference between the R2 or Adjusted R2 values for two acceptable models with differing numbers of independent variables that would favor selecting the model with the larger number of independent variables is 0.05. Please ensure that you fully understand the process for evaluating the acceptability of different models and then selecting a preferred model from amongst multiple acceptable models before attempting to apply this criterion.

• For question 14, you will need to use the regression equation that you selected as your overall preferred regression model for question 13 to calculate the predicted salary for each of the 24 employees. Keep in mind that the predicted salary value for each employee is only a point estimate. While a point estimate is a precise value, it is not necessarily an accurate value since the standard error associated with a regression model tells us there is a degree of potential error associated with using the regression model to predict salary values. In order to answer question 14 you will need to convert the point estimate for the predicted salary for each employee to an interval estimate. To calculate the interval estimate limits for each employee, simply multiply the standard error value for the preferred regression model by 1.5 and then subtract this value from the predicted point estimate salary value to define the lower limit of the interval estimate and add this value to the predicted point estimate salary value to define the upper limit for the interval estimate. Once you have calculated the interval estimate for each employee, you will then simply compare each employee's current salary to the corresponding interval estimate for their predicted salary in order to determine if his or her current salary falls within the range that the personnel director considers fair and reasonable.

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?

Use the results for all seven regression analyses, as applicable; in order to answer questions 9 through 14.

9. Which individual independent variables evidence a positive correlation with the dependent variable?

10. Which of the three pairs of independent variable evidences a degree of collinearity that should be cause for concern when performing multivariate linear regression (i.e., which pairs evidence a degree of correlation in excess of 0.5)?

11. If you were to consider only the three regression models that are based upon a single independent variable, which of the models would be your preferred model? Explain the basis for your answer.

12. If you were to consider only the three regression models that are based upon two independent variables, which of the models would be your preferred model? Explain the basis for your answer.

13. Is the regression model based upon three independent variables an acceptable model? Explain the basis for your answer.

14. When you consider all seven regression models, which of the models would be your overall preferred regression model? Explain the basis for your answer.

15. The personnel director considers an employee’s current salary to be fair and reasonable if it is within plus or minus 1.5 standard errors of the value estimated using the preferred regression model. Of the 24 employees, how many employees’ current salary is outside the range of what the personnel director considers fair and reasonable?