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
|
48.2
|
2.18
|
9
|
6
|
|
2
|
55.3
|
3.31
|
20
|
6
|
|
3
|
53.7
|
3.18
|
18
|
7
|
|
4
|
61.8
|
3.62
|
33
|
7
|
|
5
|
56.4
|
2.62
|
31
|
8
|
|
6
|
52.5
|
3.75
|
13
|
6
|
|
7
|
54.0
|
4.25
|
25
|
6
|
|
8
|
55.7
|
3.43
|
30
|
4
|
|
9
|
45.1
|
1.93
|
5
|
6
|
|
10
|
67.9
|
4.5
|
47
|
8
|
|
11
|
53.2
|
2.81
|
25
|
5
|
|
12
|
46.8
|
3.06
|
11
|
6
|
|
13
|
58.3
|
5
|
23
|
8
|
|
14
|
59.1
|
4.06
|
35
|
7
|
|
15
|
57.8
|
4.12
|
39
|
5
|
|
16
|
48.6
|
2.31
|
21
|
4
|
|
17
|
49.2
|
3.87
|
7
|
6
|
|
18
|
63.0
|
4.37
|
40
|
7
|
|
19
|
53.0
|
2.5
|
35
|
6
|
|
20
|
50.9
|
2.81
|
23
|
4
|
|
21
|
55.4
|
3.68
|
33
|
5
|
|
22
|
51.8
|
3.5
|
27
|
4
|
|
23
|
60.2
|
3
|
34
|
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. 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
Hint: 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.
Hint: For the purposes of this homework assignment, 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 0.03. Please ensure that you fully understand the process for selecting a preferred model before
attempting to apply this criterion.
Hint: Question 19 is intended to have you demonstrate that you understand how to determine which univariate models are
acceptable, and then select a preferred univariate model from amongst the acceptable univariate models. Question 20 is
intended to have you demonstrate that you understand how to determine which bivariate models are acceptable, and then
select a preferred bivariate model from amongst the acceptable bivariate models. Question 21 is intended to have you
demonstrate that you understand how to select a preferred model from amongst multiple acceptable models that have
differing numbers of independent variables. Question 22 is intended to have you demonstrate that you understand how to
determine if the trivariate model is acceptable, and then select a preferred model from amongst multiple acceptable models.
Hint: For questions 24 and 25, you need to use the regression equation associated with the preferred model selected for
question 22 in order to calculate the predicted salary for each of the 24 employees. In order to answer questions 24 and 25
you need to keep in mind that the predicted salary value for each employee is only a point estimate (this concept was
discussed in week one relative to the mean). While a point estimate is a precise value, it is not necessarily an accurate value
since the standard error value tells us there is some potential degree of error associated with using the preferred regression
model to predict salary values. In order to answer questions 24 and 25 you will need to create an interval estimate (this
concept was also discussed during week one relative to the mean) for the predicted salary for each of the 24 employees. To
calculate the interval estimate 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 created an interval estimate for each employee, you will then need to compare each employee's current salary
to their corresponding interval estimate in order to determine if each employee's current salary falls within their predicted
interval estimate.
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 14.
1. What is the degree of correlation between the dependent variable and the independent variable?
2. Does the regression model confirm a positive correlation between the dependent variable and the independent variable as
hypothesized?
3. What is the desired statistical significance for the regression model?
4. Is the statistical significance of the model as a whole less than the desired statistical significance for the regression model?
5. What is the actual confidence level for the regression model as a whole?
6. What is the actual confidence level for the regression model as a whole?
7. Is the statistical significance of the linear relationship between the dependent and independent variables less than the desired statistical significance for the regression model?
8. Should the coefficient of determination or adjusted coefficient of determination be used to evaluate this regression model?
- coefficient of determination
- adjusted coefficient of determination
9. 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?
10. What is the amount by which we will be off on average when predicting values for the dependent variable using the regression model?
11. What is the coefficient for the y-intercept for the regression model?
12. What is the coefficient for the y-intercept for the regression model?
13. What is the coefficient for the independent variable for the regression model?
14. What is the point estimate for the predicted salary for an employee with an average performance rating of 3.9?
15. What is the interval estimate for the predicted salary for an employee with an average performance rating of 3.9 based
upon taking into consideration the standard error?
16. What is the 95% confidence level interval estimate for the salary for an employee with an average performance rating of
3.9?
Perform a correlation analysis between the dependent variable and each of the three independent variables. Use the results of
the correlation analysis to answer questions 15 and 16.
1. Which independent variables evidence a positive correlation with the dependent variable?
- Average performance rating for the past 3 years
- Number of relevant certifications
Perform a correlation analysis between each of the three pairs of independent variables. Use the results of the correlation
analyses to answer question 17.
1. Which pair of independent variable evidences a degree of collinearity that should be cause for concern when performing
multivariate linear regression (i.e., evidences a degree of correlation in excess of 0.5)?
- Average performance rating for the past 3 years vs. number of relevant certifications
- Years of service vs. number of relevant certifications
- Average performance rating for the past 3 years vs. years of service
Use the regression statistics pertaining to all seven regression analyses in order to answer questions 15 through 23.
2. Of the seven regression models, which model both accounts for the lowest 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 and evidences
the highest degree of error for predicting values for the dependent variable?
- 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
3. If you were to consider only the three regression models that are based upon a single independent variable, which of the
following models would be your preferred model?
- Annual salary vs. average performance rating for the past 3 years
- Annual salary vs. years of service
- Annual salary vs. number of relevant certifications
4. If you were to compare the preferred regression model based upon a single independent variable with the preferred
regression model based upon two independent variables, which model would be preferred overall?
- Preferred regression model based upon a single independent variable
- Preferred regression model based upon two independent variables
5. When you consider all seven regression models, which is the overall preferred regression model?
- 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
6. Do any of the regression models offer a higher confidence level for the model as a whole, or a lower standard error in
comparison to the overall preferred model?
The personnel is interested in comparing each employee's actual salary to their predicted salary in order to determine if there
are any prevailing salary inequities. Suppose 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 by the regression model selected in response
to question 22. For each individual employee, calculate his or her estimated salary using the regression model selected in
response to question 22, as well calculate his or her upper and lower limits for a fair reasonable salary, in order to answer
questions 24 and 25.
1. Of the 24 employees, how many employees' current salary is below what is considered fair and reasonable?
2. Of the 24 employees, how many employees' current salary is above what is considered fair and reasonable?