Statistics Week 4

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Data

See comments at the right of the data set.
ID Salary Compa Midpoint Age Performance Rating Service Gender Raise Degree Gender1 Grade
8 23 1.000 23 32 90 9 1 5.8 0 F A The ongoing question that the weekly assignments will focus on is: Are males and females paid the same for equal work (under the Equal Pay Act)?
10 22 0.956 23 30 80 7 1 4.7 0 F A Note: to simplfy the analysis, we will assume that jobs within each grade comprise equal work.
11 23 1.000 23 41 100 19 1 4.8 0 F A
14 24 1.043 23 32 90 12 1 6 0 F A The column labels in the table mean:
15 24 1.043 23 32 80 8 1 4.9 0 F A ID – Employee sample number Salary – Salary in thousands
23 23 1.000 23 36 65 6 1 3.3 1 F A Age – Age in years Performance Rating – Appraisal rating (Employee evaluation score)
26 24 1.043 23 22 95 2 1 6.2 1 F A Service – Years of service (rounded) Gender: 0 = male, 1 = female
31 24 1.043 23 29 60 4 1 3.9 0 F A Midpoint – salary grade midpoint Raise – percent of last raise
35 24 1.043 23 23 90 4 1 5.3 1 F A Grade – job/pay grade Degree (0= BS\BA 1 = MS)
36 23 1.000 23 27 75 3 1 4.3 1 F A Gender1 (Male or Female) Compa - salary divided by midpoint
37 22 0.956 23 22 95 2 1 6.2 1 F A
42 24 1.043 23 32 100 8 1 5.7 0 F A
3 34 1.096 31 30 75 5 1 3.6 0 F B
18 36 1.161 31 31 80 11 1 5.6 1 F B
20 34 1.096 31 44 70 16 1 4.8 1 F B
39 35 1.129 31 27 90 6 1 5.5 1 F B
7 41 1.025 40 32 100 8 1 5.7 0 F C 84.2
13 42 1.050 40 30 100 2 1 4.7 1 F C 87.6
22 57 1.187 48 48 65 6 1 3.8 0 F D
24 50 1.041 48 30 75 9 1 3.8 1 F D
45 55 1.145 48 36 95 8 1 5.2 0 F D
17 69 1.210 57 27 55 3 1 3 0 F E
48 65 1.140 57 34 90 11 1 5.3 1 F E
28 75 1.119 67 44 95 9 1 4.4 1 F F
43 77 1.149 67 42 95 20 1 5.5 1 F F
19 24 1.043 23 32 85 1 0 4.6 1 M A
25 24 1.043 23 41 70 4 0 4 0 M A
40 25 1.086 23 24 90 2 0 6.3 0 M A
2 27 0.870 31 52 80 7 0 3.9 0 M B
32 28 0.903 31 25 95 4 0 5.6 0 M B
34 28 0.903 31 26 80 2 0 4.9 1 M B
16 47 1.175 40 44 90 4 0 5.7 0 M C
27 40 1.000 40 35 80 7 0 3.9 1 M C
41 43 1.075 40 25 80 5 0 4.3 0 M C
5 47 0.979 48 36 90 16 0 5.7 1 M D
30 49 1.020 48 45 90 18 0 4.3 0 M D
1 58 1.017 57 34 85 8 0 5.7 0 M E
4 66 1.157 57 42 100 16 0 5.5 1 M E
12 60 1.052 57 52 95 22 0 4.5 0 M E
33 64 1.122 57 35 90 9 0 5.5 1 M E
38 56 0.982 57 45 95 11 0 4.5 0 M E
44 60 1.052 57 45 90 16 0 5.2 1 M E
46 65 1.140 57 39 75 20 0 3.9 1 M E
47 62 1.087 57 37 95 5 0 5.5 1 M E
49 60 1.052 57 41 95 21 0 6.6 0 M E
50 66 1.157 57 38 80 12 0 4.6 0 M E
6 76 1.134 67 36 70 12 0 4.5 1 M F
9 77 1.149 67 49 100 10 0 4 1 M F
21 76 1.134 67 43 95 13 0 6.3 1 M F
29 72 1.074 67 52 95 5 0 5.4 0 M F

Week 1

Week 1. Measurement and Description - chapters 1 and 2
1 Measurement issues. Data, even numerically coded variables, can be one of 4 levels -
nominal, ordinal, interval, or ratio. It is important to identify which level a variable is, as
this impact the kind of analysis we can do with the data. For example, descriptive statistics
such as means can only be done on interval or ratio level data.
Please list under each label, the variables in our data set that belong in each group.
Nominal Ordinal Interval Ratio
b. For each variable that you did not call ratio, why did you make that decision?
2 The first step in analyzing data sets is to find some summary descriptive statistics for key variables.
For salary, compa, age, performance rating, and service; find the mean, standard deviation, and range for 3 groups: overall sample, Females, and Males.
You can use either the Data Analysis Descriptive Statistics tool or the Fx =average and =stdev functions.
(the range must be found using the difference between the =max and =min functions with Fx) functions.
Note: Place data to the right, if you use Descriptive statistics, place that to the right as well.
Salary Compa Age Perf. Rat. Service
Overall Mean
Standard Deviation
Range
Female Mean
Standard Deviation
Range
Male Mean
Standard Deviation
Range
3 What is the probability for a: Probability
a.       Randomly selected person being a male in grade E?
b.      Randomly selected male being in grade E?
Note part b is the same as given a male, what is probabilty of being in grade E?
c. Why are the results different?
4 For each group (overall, females, and males) find: Overall Female Male
a. The value that cuts off the top 1/3 salary in each group.
b. The z score for each value:
c. The normal curve probability of exceeding this score:
d. What is the empirical probability of being at or exceeding this salary value?
e. The value that cuts off the top 1/3 compa in each group.
f. The z score for each value:
g. The normal curve probability of exceeding this score:
h. What is the empirical probability of being at or exceeding this compa value?
i. How do you interpret the relationship between the data sets? What do they mean about our equal pay for equal work question?
5.      What conclusions can you make about the issue of male and female pay equality? Are all of the results consistent?
What is the difference between the sal and compa measures of pay?
Conclusions from looking at salary results:
Conclusions from looking at compa results:
Do both salary measures show the same results?
Can we make any conclusions about equal pay for equal work yet?

Week 2

Week 2 Testing means Q3
In questions 2 and 3, be sure to include the null and alternate hypotheses you will be testing. Ho Female Male Female
In the first 3 questions use alpha = 0.05 in making your decisions on rejecting or not rejecting the null hypothesis. 45 34 1.017 1.096
45 41 0.870 1.025
1 Below are 2 one-sample t-tests comparing male and female average salaries to the overall sample mean. 45 23 1.157 1.000
(Note: a one-sample t-test in Excel can be performed by selecting the 2-sample unequal variance t-test and making the second variable = Ho value -- see column S) 45 22 0.979 0.956
Based on our sample, how do you interpret the results and what do these results suggest about the population means for male and female average salaries? 45 23 1.134 1.000
Males Females 45 42 1.149 1.050
Ho: Mean salary = 45 Ho: Mean salary = 45 45 24 1.052 1.043
Ha: Mean salary =/= 45 Ha: Mean salary =/= 45 45 24 1.175 1.043
45 69 1.043 1.210
Note: While the results both below are actually from Excel's t-Test: Two-Sample Assuming Unequal Variances, 45 36 1.134 1.161
having no variance in the Ho variable makes the calculations default to the one-sample t-test outcome - we are tricking Excel into doing a one sample test for us. 45 34 1.043 1.096
Male Ho Female Ho 45 57 1.000 1.187
Mean 52 45 Mean 38 45 45 23 1.074 1.000
Variance 316 0 Variance 334.6666666667 0 45 50 1.020 1.041
Observations 25 25 Observations 25 25 45 24 0.903 1.043
Hypothesized Mean Difference 0 Hypothesized Mean Difference 0 45 75 1.122 1.119
df 24 df 24 45 24 0.903 1.043
t Stat 1.9689038266 t Stat -1.9132063573 45 24 0.982 1.043
P(T<=t) one-tail 0.0303078503 P(T<=t) one-tail 0.0338621184 45 23 1.086 1.000
t Critical one-tail 1.7108820799 t Critical one-tail 1.7108820799 45 22 1.075 0.956
P(T<=t) two-tail 0.0606157006 P(T<=t) two-tail 0.0677242369 45 35 1.052 1.129
t Critical two-tail 2.0638985616 t Critical two-tail 2.0638985616 45 24 1.140 1.043
Conclusion: Do not reject Ho; mean equals 45 Conclusion: Do not reject Ho; mean equals 45 4.7733406049 45 77 1.087 1.149
Is this a 1 or 2 tail test? 2-tail Is this a 1 or 2 tail test? 2-tail
- why? Because we are testing whether the salary is either larger or smaller than the hypothesis value, meaning the test is done on both directions of the mean - why? Because we are testing whether the salary is either larger or smaller than the hypothesis value, meaning the test is done on both directions of the mean
P-value is: 0.0546333775 P-value is: 0.0615740746 45 55 1.052 1.145
Is P-value > 0.05? Yes Is P-value > 0.05? Yes 45 65 1.157 1.140
Why do we not reject Ho? Because the p-value is greater than 0.05, indicating weak evidence against the null hypothesis Why do we not reject Ho? Because the p-value is greater than 0.05, indicating weak evidence against the null hypothesis
Interpretation: The data suggests that male employee salaries are generally higher than female employee salaries for employees
2 Based on our sample data set, perform a 2-sample t-test to see if the population male and female average salaries could be equal to each other.
(Since we have not yet covered testing for variance equality, assume the data sets have statistically equal variances.)
Ho: 45
Ha: 60
Test to use: 2-sample equal variance t-test
Place B43 in Outcome range box.
0.0001086152
P-value is: 0.00010862
Is P-value < 0.05? No
Reject or do not reject Ho: Do not reject
If the null hypothesis was rejected, what is the effect size value:
Meaning of effect size measure: It shows the degree of inequality between male and female salaries
Interpretation:
b. Since the one and two tail t-test results provided different outcomes, which is the proper/correct apporach to comparing salary equality? Why?
The two-tail approach is correct in this situation because it considers both the possibility of males being paid higher than females as well as the vice versa situation.
3 Based on our sample data set, can the male and female compas in the population be equal to each other? (Another 2-sample t-test.)
Ho: 1
Ha: 1.5
Statistical test to use: 2-sample equal variance t-test
Place B75 in Outcome range box.
What is the p-value: 1.8760E-14
Is P-value < 0.05? Yes
Reject or do not reject Ho: Reject
If the null hypothesis was rejected, what is the effect size value: 5.3063E-15
Meaning of effect size measure: It shows that the comparative salaries between males and females are considerably different, with the males tipping the scales on the higher side.
Interpretation: More men are generally paid salaries that are higher than the collective average as compared to women.
4 Since performance is often a factor in pay levels, is the average Performance Rating the same for both genders?
Ho: 90
Ha: 85
Test to use: 2-sample equal variance t-test
Place B106 in Outcome range box.
0.2654912702
What is the p-value: 0.2654912702
Is P-value < 0.05? No
Do we REJ or Not reject the null? We do not reject the null hypothesis
If the null hypothesis was rejected, what is the effect size value:
Meaning of effect size measure: Performance for men is higher than for women by a slight margin
Interpretation: Performance could be correctly considered as one of the factors contributing to unequal pay between men and women. However, the difference in performance between the genders is not very large.
5 If the salary and compa mean tests in questions 2 and 3 provide different results about male and female salary equality,
which would be more appropriate to use in answering the question about salary equity? Why?
What are your conclusions about equal pay at this point?
The compa test is more suitable for use in comparison in this situation. This is because each employee's salary is weighed against the company average before the data is used further. This makes it a more useful measure as it gives an indication of the existent inequality or equality when measured against a fair and reasonable standard.
I conclude that the salaries of males are higher than those of females. This is the case in the given scenario as well as in the real-world job environment.

Week 3

Week 3
At this point we know the following about male and female salaries.
a. Male and female overall average salaries are not equal in the population.
b. Male and female overall average compas are equal in the population, but males are a bit more spread out.
c. The male and female salary range are almost the same, as is their age and service.
d. Average performance ratings per gender are equal.
Let's look at some other factors that might influence pay - education(degree) and performance ratings.
1 Last week, we found that average performance ratings do not differ between males and females in the population.
Now we need to see if they differ among the grades. Is the average performace rating the same for all grades?
(Assume variances are equal across the grades for this ANOVA.) A B C D E F
Null Hypothesis:
Alt. Hypothesis:
Place B17 in Outcome range box.
Interpretation:
What is the p-value:
Is P-value < 0.05?
Do we REJ or Not reject the null?
If the null hypothesis was rejected, what is the effect size value (eta squared):
Meaning of effect size measure:
What does that decision mean in terms of our equal pay question:
2 While it appears that average salaries per each grade differ, we need to test this assumption.
Is the average salary the same for each of the grade levels? (Assume equal variance, and use the analysis toolpak function ANOVA.)
Use the input table to the right to list salaries under each grade level.
Null Hypothesis:
Alt. Hypothesis: A B C D E F
Place B55 in Outcome range box.
What is the p-value:
Is P-value < 0.05?
Do you reject or not reject the null hypothesis:
If the null hypothesis was rejected, what is the effect size value (eta squared):
Meaning of effect size measure:
Interpretation:
3 The table and analysis below demonstrate a 2-way ANOVA with replication. Please interpret the results.
BA MA Ho: Average compas by gender are equal
Male 1.017 1.157 Ha: Average compas by gender are not equal
0.870 0.979 Ho: Average compas are equal for each degree
1.052 1.134 Ho: Average compas are not equal for each degree
1.175 1.149 Ho: Interaction is not significant
1.043 1.043 Ha: Interaction is significant
1.074 1.134
1.020 1.000 Perform analysis:
0.903 1.122
0.982 0.903 Anova: Two-Factor With Replication
1.086 1.052
1.075 1.140 SUMMARY BA MA Total
1.052 1.087 Male
Female 1.096 1.050 Count 12 12 24
1.025 1.161 Sum 12.349 12.9 25.249
1.000 1.096 Average 1.0290833333 1.075 1.0520416667
0.956 1.000 Variance 0.006686447 0.0065198182 0.0068660417
1.000 1.041
1.043 1.043 Female
1.043 1.119 Count 12 12 24
1.210 1.043 Sum 12.791 12.787 25.578
1.187 1.000 Average 1.0659166667 1.0655833333 1.06575
1.043 0.956 Variance 0.006102447 0.0042128106 0.004933413
1.043 1.129
1.145 1.149 Total
Count 24 24
Sum 25.14 25.687
Average 1.0475 1.0702916667
Variance 0.0064703478 0.0051561286
ANOVA
Source of Variation SS df MS F P-value F crit
Sample 0.0022550208 1 0.0022550208 0.3834821171 0.5389389507 4.0617064601 (This is the row variable or gender.)
Columns 0.0062335208 1 0.0062335208 1.0600539609 0.3088295633 4.0617064601 (This is the column variable or Degree.)
Interaction 0.0064171875 1 0.0064171875 1.0912877664 0.3018915062 4.0617064601
Within 0.25873675 44 0.0058803807
Total 0.2736424792 47
Interpretation:
For Ho: Average compas by gender are equal Ha: Average compas by gender are not equal
What is the p-value:
Is P-value < 0.05?
Do you reject or not reject the null hypothesis:
If the null hypothesis was rejected, what is the effect size value (eta squared):
Meaning of effect size measure:
For Ho: Average salaries are equal for all grades Ha: Average salaries are not equal for all grades
What is the p-value:
Is P-value < 0.05?
Do you reject or not reject the null hypothesis:
If the null hypothesis was rejected, what is the effect size value (eta squared):
Meaning of effect size measure:
For: Ho: Interaction is not significant Ha: Interaction is significant
What is the p-value:
Do you reject or not reject the null hypothesis:
If the null hypothesis was rejected, what is the effect size value (eta squared):
Meaning of effect size measure:
What do these decisions mean in terms of our equal pay question:
4 Many companies consider the grade midpoint to be the "market rate" - what is needed to hire a new employee. Midpoint Salary
Does the company, on average, pay its existing employees at or above the market rate?
Null Hypothesis:
Alt. Hypothesis:
Statistical test to use:
Place the cursor in B160 for correl.
What is the p-value:
Is P-value < 0.05?
Do we REJ or Not reject the null?
If the null hypothesis was rejected, what is the effect size value: Since the effect size was not discussed in this chapter, we do not have a formula for it - it differs from the non-paired t.
Meaning of effect size measure: NA
Interpretation:
5.   Using the results up thru this week, what are your conclusions about gender equal pay for equal work at this point?

Week 4

Week 4 Confidence Intervals and Chi Square (Chs 11 - 12)
For questions 3 and 4 below, be sure to list the null and alternate hypothesis statements. Use .05 for your significance level in making your decisions.
For full credit, you need to also show the statistical outcomes - either the Excel test result or the calculations you performed.
1 Using our sample data, construct a 95% confidence interval for the population's mean salary for each gender.
Interpret the results. How do they compare with the findings in the week 2 one sample t-test outcomes (Question 1)?
Mean St error t value Low to High
Males
Females
<Reminder: standard error is the sample standard deviation divided by the square root of the sample size.>
Interpretation:
2 Using our sample data, construct a 95% confidence interval for the mean salary difference between the genders in the population.
How does this compare to the findings in week 2, question 2?
Difference St Err. T value Low to High
Yes/No
Can the means be equal? Why?
How does this compare to the week 2, question 2 result (2 sampe t-test)?
a. Why is using a two sample tool (t-test, confidence interval) a better choice than using 2 one-sample techniques when comparing two samples?
3 We found last week that the degrees compa values within the population.
do not impact compa rates. This does not mean that degrees are distributed evenly across the grades and genders.
Do males and females have athe same distribution of degrees by grade?
(Note: while technically the sample size might not be large enough to perform this test, ignore this limitation for this exercise.)
What are the hypothesis statements:
Ho:
Ha:
Note: You can either use the Excel Chi-related functions or do the calculations manually.
Data input tables - graduate degrees by gender and grade level
OBSERVED A B C D E F Total Do manual calculations per cell here (if desired)
M Grad A B C D E F
Fem Grad M Grad
Male Und Fem Grad
Female Und Male Und
Female Und
Sum =
EXPECTED
M Grad For this exercise - ignore the requirement for a correction
Fem Grad for expected values less than 5.
Male Und
Female Und
Interpretation:
What is the value of the chi square statistic:
What is the p-value associated with this value:
Is the p-value <0.05?
Do you reject or not reject the null hypothesis:
If you rejected the null, what is the Cramer's V correlation:
What does this correlation mean?
What does this decision mean for our equal pay question:
4 Based on our sample data, can we conclude that males and females are distributed across grades in a similar pattern
within the population?
What are the hypothesis statements:
Ho:
Ha:
Do manual calculations per cell here (if desired)
A B C D E F A B C D E F
OBS COUNT - m M
OBS COUNT - f F
Sum =
EXPECTED
What is the value of the chi square statistic:
What is the p-value associated with this value:
Is the p-value <0.05?
Do you reject or not reject the null hypothesis:
If you rejected the null, what is the Phi correlation:
What does this correlation mean?
What does this decision mean for our equal pay question:
5.      How do you interpret these results in light of our question about equal pay for equal work?

Week 5

Week 5 Correlation and Regression
1.     Create a correlation table for the variables in our data set. (Use analysis ToolPak or StatPlus:mac LE function Correlation.)
a. Reviewing the data levels from week 1, what variables can be used in a Pearson's Correlation table (which is what Excel produces)?
b. Place table here (C8 in Output range box):
c. Using r = approximately .28 as the signicant r value (at p = 0.05) for a correlation between 50 values, what variables are
significantly related to Salary?
To compa?
d. Looking at the above correlations - both significant or not - are there any surprises -by that I
mean any relationships you expected to be meaningful and are not and vice-versa?
e. Does this help us answer our equal pay for equal work question?
2 Below is a regression analysis for salary being predicted/explained by the other variables in our sample (Midpoint,
age, performance rating, service, gender, and degree variables. (Note: since salary and compa are different ways of
expressing an employee’s salary, we do not want to have both used in the same regression.)
Plase interpret the findings.
Ho: The regression equation is not significant.
Ha: The regression equation is significant.
Ho: The regression coefficient for each variable is not significant Note: technically we have one for each input variable.
Ha: The regression coefficient for each variable is significant Listing it this way to save space.
Sal
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.9915590747
R Square 0.9831893985
Adjusted R Square 0.9808437332
Standard Error 2.6575925726
Observations 50
ANOVA
df SS MS F Significance F
Regression 6 17762.2996738743 2960.383278979 419.1516111294 1.8121523852609E-36
Residual 43 303.7003261257 7.062798282
Total 49 18066
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept -1.7496212123 3.6183676583 -0.4835388157 0.6311664899 -9.0467550427 5.547512618 -9.0467550427 5.547512618
Midpoint 1.2167010505 0.0319023509 38.1382881163 8.66416336978111E-35 1.1523638283 1.2810382727 1.1523638283 1.2810382727
Age -0.0046280102 0.065197212 -0.0709847876 0.9437389875 -0.1361107191 0.1268546987 -0.1361107191 0.1268546987
Performace Rating -0.0565964405 0.0344950678 -1.6407110971 0.1081531819 -0.1261623747 0.0129694936 -0.1261623747 0.0129694936
Service -0.0425003573 0.0843369821 -0.5039350033 0.6168793519 -0.2125820912 0.1275813765 -0.2125820912 0.1275813765
Gender 2.420337212 0.8608443176 2.8115852804 0.0073966188 0.684279192 4.156395232 0.684279192 4.156395232
Degree 0.2755334143 0.7998023048 0.3445019009 0.732148119 -1.3374216547 1.8884884833 -1.3374216547 1.8884884833
Note: since Gender and Degree are expressed as 0 and 1, they are considered dummy variables and can be used in a multiple regression equation.
Interpretation:
For the Regression as a whole:
What is the value of the F statistic:
What is the p-value associated with this value:
Is the p-value <0.05?
Do you reject or not reject the null hypothesis:
What does this decision mean for our equal pay question:
For each of the coefficients: Intercept Midpoint Age Perf. Rat. Service Gender Degree
What is the coefficient's p-value for each of the variables:
Is the p-value < 0.05?
Do you reject or not reject each null hypothesis:
What are the coefficients for the significant variables?
Using only the significant variables, what is the equation? Salary =
Is gender a significant factor in salary:
If so, who gets paid more with all other things being equal?
How do we know?
3 Perform a regression analysis using compa as the dependent variable and the same independent
variables as used in question 2. Show the result, and interpret your findings by answering the same questions.
Note: be sure to include the appropriate hypothesis statements.
Regression hypotheses
Ho:
Ha:
Coefficient hypotheses (one to stand for all the separate variables)
Ho:
Ha:
Put C94 in output range box
Interpretation:
For the Regression as a whole:
What is the value of the F statistic:
What is the p-value associated with this value:
Is the p-value < 0.05?
Do you reject or not reject the null hypothesis:
What does this decision mean for our equal pay question:
For each of the coefficients: Intercept Midpoint Age Perf. Rat. Service Gender Degree
What is the coefficient's p-value for each of the variables:
Is the p-value < 0.05?
Do you reject or not reject each null hypothesis:
What are the coefficients for the significant variables?
Using only the significant variables, what is the equation? Compa =
Is gender a significant factor in compa:
If so, who gets paid more with all other things being equal?
How do we know?
4 Based on all of your results to date, do we have an answer to the question of are males and females paid equally for equal work?
If so, which gender gets paid more?
How do we know?
Which is the best variable to use in analyzing pay practices - salary or compa? Why?
What is most interesting or surprising about the results we got doing the analysis during the last 5 weeks?
5 Why did the single factor tests and analysis (such as t and single factor ANOVA tests on salary equality) not provide a complete answer to our salary equality question?
What outcomes in your life or work might benefit from a multiple regression examination rather than a simpler one variable test?