STATS Week Three Homework Assignment – Linear Regression

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. 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

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 R2or
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?
o 0.8198
o 0.6672
o 0.7862
o 0.6523

2. Does
the regression model confirm a positive correlation between the dependent
variable and the independent variable as hypothesized?
o Yes
o No

3. What
is the desired statistical significance for the regression model?
o 0.00
o 0.01
o 0.05
o 0.10

4. Is
the statistical significance of the model as a whole less than the desired
statistical significance for the regression model?
o Yes
o No

5. What
is the actual confidence level for the regression model as a whole?
o 99.97%
o 95%
o 90.5%
o 80.2%

6. Is
the statistical significance of the linear relationship between the dependent
and independent variables less than the desired statistical significance for
the regression model?
o Yes
o No

7. Should
the coefficient of determination or adjusted coefficient of determination be
used to evaluate this regression model?
o Coefficient
of determination
o Adjusted
coefficient of determination

8. 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?
o 44.51%
o 66.72%
o 41.99%
o 76.24%

9. What
is the amount by which we will be off on average when predicting values for the
dependent variable using the regression model?
o $12,287
o $32,966
o $4,169
o $25,896

10. What is the
coefficient for the y-intercept for the regression model?
o 39.38
o 19.94
o 12.29
o 2.96

11. What is the
coefficient for the independent variable for the regression model?
o 63.08
o 4.52
o 12.29
o 2.96

12. What is the
point estimate for the predicted salary for an employee with an average
performance rating of 3.9?
o $71,562
o $57,006
o $41,299
o $50,896

13. 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?
o $61,913
– $71,562
o $57,006
– $64,159
o $52,837
– $61,175
o $54,896
– $60,873

14. What is the
95% confidence level interval estimate for the salary for an employee with an
average performance rating of 3.9?
o $60,265
– 85,789
o $75,412
– $78,523
o $40,636
– $73,376
o $65,141
– $72,269

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.

15. Which
independent variables evidence a positive correlation with the dependent
variable?
o
Average
performance rating for the past 3 years
o
Years
of service
o
Number
of relevant certifications
o All
of the above
o None
of the above

16. Which
independent variable evidences the highest degree of correlation with the
dependent variable?
o
Average
performance rating for the past 3 years
o
Years
of service
o
Number
of relevant certifications
o All
of the above
o None
of the above

Perform a correlation
analysis between each of the three pairs of independent variables. Use the results of the correlation analyses
to answer question 17.

17. 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)?
o
Average
performance rating for the past 3 years vs. years of service
o
Average
performance rating for the past 3 years vs. number of relevant certifications
o
Years
of service vs. number of relevant certifications
o All
of the above
o None
of the above

Use the regression
statistics pertaining to all seven regression analyses in order to answer
questions 15 through 23.

18.
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?
o
Annual
salary vs. average performance rating for the past 3 years
o
Annual
salary vs. years of service
o
Annual
salary vs. number of relevant certifications
o
Annual
salary vs. average performance rating for the past 3 years and years of service
o
Annual
salary vs. average performance rating for the past 3 years and number of
relevant certifications
o
Annual
salary vs. years of service and number of relevant certifications
o
Annual
salary vs. average performance rating for the past 3 years, years of service
and number of relevant certifications

19.
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?
o
Annual
salary vs. average performance rating for the past 3 years
o
Annual
salary vs. years of service
o
Annual
salary vs. number of relevant certifications

20.
If
you were to consider only the three regression models that are based upon two
independent variables, which of the following models would be your preferred
model?
o
Annual
salary vs. average performance rating for the past 3 years and years of service
o
Annual
salary vs. average performance rating for the past 3 years and number of
relevant certifications
o
Annual
salary vs. years of service and number of relevant certifications

21.
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?
o
Preferred
regression model based upon a single independent variable
o
Preferred
regression model based upon two independent variables

22.
When
you consider all seven regression models, which is the overall preferred regression
model?
o
Annual
salary vs. average performance rating for the past 3 years
o
Annual
salary vs. years of service
o
Annual
salary vs. number of relevant certifications
o
Annual
salary vs. average performance rating for the past 3 years and years of service
o
Annual
salary vs. average performance rating for the past 3 years and number of
relevant certifications
o
Annual
salary vs. years of service and number of relevant certifications
o
Annual
salary vs. average performance rating for the past 3 years, years of service
and number of relevant certifications

23.
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?
o
Yes
o
No

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.

24.
Of
the 24 employees, how many employees’ current salary is below what is
considered fair and reasonable?
o
0
o
1
o
2
o
3
o
4
o
5

25.
Of
the 24 employees, how many employees’ current salary is above what is
considered fair and reasonable?
o
0
o
1
o
2
o
3
o
4
o
5