BUS308: Statistics for Managers

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week_3_homework.xlsx
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Data

ID Salary Compa Midpoint Age Performance Rating Service Gender Raise Degree Gender1 Gr Students: Copy the Student Data file data values into this sheet to assist in doing your weekly assignments.
1 60.3 1.058 57 34 85 8 0 5.7 0 M E 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)?
2 27.2 0.878 31 52 80 7 0 3.9 0 M B Note: to simplfy the analysis, we will assume that jobs within each grade comprise equal work.
3 35 1.129 31 30 75 5 1 3.6 1 F B
4 61 1.070 57 42 100 16 0 5.5 1 M E The column labels in the table mean:
5 47.7 0.993 48 36 90 16 0 5.7 1 M D ID – Employee sample number Salary – Salary in thousands
6 76 1.135 67 36 70 12 0 4.5 1 M F Age – Age in years Performance Rating - Appraisal rating (employee evaluation score)
7 41 1.026 40 32 100 8 1 5.7 1 F C Service – Years of service (rounded) Gender – 0 = male, 1 = female
8 21.5 0.933 23 32 90 9 1 5.8 1 F A Midpoint – salary grade midpoint Raise – percent of last raise
9 74.3 1.109 67 49 100 10 0 4 1 M F Grade – job/pay grade Degree (0= BS\BA 1 = MS)
10 23.6 1.025 23 30 80 7 1 4.7 1 F A Gender1 (Male or Female) Compa - salary divided by midpoint
11 24.8 1.078 23 41 100 19 1 4.8 1 F A
12 66.1 1.159 57 52 95 22 0 4.5 0 M E
13 41 1.026 40 30 100 2 1 4.7 0 F C
14 22.1 0.963 23 32 90 12 1 6 1 F A
15 24.6 1.069 23 32 80 8 1 4.9 1 F A
16 47.1 1.177 40 44 90 4 0 5.7 0 M C
17 69.6 1.221 57 27 55 3 1 3 1 F E
18 35 1.130 31 31 80 11 1 5.6 0 F B
19 24.9 1.081 23 32 85 1 0 4.6 1 M A
20 34.2 1.104 31 44 70 16 1 4.8 0 F B
21 74.4 1.110 67 43 95 13 0 6.3 1 M F
22 51.4 1.070 48 48 65 6 1 3.8 1 F D
23 23.4 1.018 23 36 65 6 1 3.3 0 F A
24 56.4 1.176 48 30 75 9 1 3.8 0 F D
25 25.1 1.093 23 41 70 4 0 4 0 M A
26 23 0.999 23 22 95 2 1 6.2 0 F A
27 42.5 1.063 40 35 80 7 0 3.9 1 M C
28 75 1.120 67 44 95 9 1 4.4 0 F F
29 79.9 1.193 67 52 95 5 0 5.4 0 M F
30 48.8 1.017 48 45 90 18 0 4.3 0 M D
31 24.1 1.047 23 29 60 4 1 3.9 1 F A
32 27.4 0.885 31 25 95 4 0 5.6 0 M B
33 60.6 1.063 57 35 90 9 0 5.5 1 M E
34 27.3 0.882 31 26 80 2 0 4.9 1 M B
35 24.4 1.062 23 23 90 4 1 5.3 0 F A
36 22.7 0.985 23 27 75 3 1 4.3 0 F A
37 22.5 0.977 23 22 95 2 1 6.2 0 F A
38 58.9 1.033 57 45 95 11 0 4.5 0 M E
39 34.5 1.114 31 27 90 6 1 5.5 0 F B
40 24.3 1.055 23 24 90 2 0 6.3 0 M A
41 42.6 1.066 40 25 80 5 0 4.3 0 M C
42 24.8 1.078 23 32 100 8 1 5.7 1 F A
43 75.7 1.130 67 42 95 20 1 5.5 0 F F
44 64.4 1.130 57 45 90 16 0 5.2 1 M E
45 48 1.000 48 36 95 8 1 5.2 1 F D
46 61.3 1.076 57 39 75 20 0 3.9 1 M E
47 62.9 1.103 57 37 95 5 0 5.5 1 M E
48 66.3 1.163 57 34 90 11 1 5.3 1 F E
49 60 1.053 57 41 95 21 0 6.6 0 M E
50 67.2 1.179 57 38 80 12 0 4.6 0 M E

Week 3

Week 3 Paired T-test and ANOVA
For this week's work, again be sure to state the null and alternate hypotheses and use alpha = 0.05 for our decision
value in the reject or do not reject decision on the null hypothesis.
1 Many companies consider the grade midpoint to be the "market rate" - the salary needed to hire a new employee. Salary Midpoint Diff
Does the company, on average, pay its existing employees at or above the market rate? 60.3 57 3.3
Use the data columns at the right to set up the paired data set for the analysis. 27.2 31 -3.8
35 31 4
Null Hypothesis: Difference between Salary and midpoint is 0 61 57 4
Alt. Hypothesis: Difference between Salary and midpoint is not 0 47.7 48 -0.3
76 67 9
Statistical test to use: Paired T-test 41 40 1
21.5 23 -1.5
t-Test: Paired Two Sample for Means 74.3 67 7.3
23.6 23 0.6
Salary Midpoint 24.8 23 1.8
Mean 45.136 41.76 66.1 57 9.1
Variance 373.1007183673 263.4514285714 41 40 1
Observations 50 50 22.1 23 -0.9
Pearson Correlation 0.9895407199 24.6 23 1.6
Hypothesized Mean Difference 0 47.1 40 7.1
df 49 69.6 57 12.6
t Stat 5.9543595083 35 31 4
P(T<=t) one-tail 0.0000001376 24.9 23 1.9
t Critical one-tail 1.6765508926 34.2 31 3.2
P(T<=t) two-tail 0.0000002752 74.4 67 7.4
t Critical two-tail 2.0095752371 51.4 48 3.4
23.4 23 0.4
56.4 48 8.4
25.1 23 2.1
23 23 0
What is the p-value: 2.75E-07 42.5 40 2.5
Is P-value < 0.05 (one tail test) or 0.25 (two tail test)? Yes 75 67 8
What else needs to be checked on a 1-tail test in order to reject the null? 79.9 67 12.9
Do we REJ or Not reject the null? Reject Null hypotheis 48.8 48 0.8
If the null hypothesis was rejected, what is the effect size value: 0.8420735972 24.1 23 1.1
If calculated, what is the meaning of effect size measure: The effect size is quite large 27.4 31 -3.6
60.6 57 3.6
Interpretation of test results: There is a difference between Salary and midpoint and the 84% variability is explained by the treatment effect. 27.3 31 -3.7
24.4 23 1.4
22.7 23 -0.3
22.5 23 -0.5
Let's look at some other factors that might influence pay - education(degree) and performance ratings. 58.9 57 1.9
34.5 31 3.5
2 Last week, we found that average performance ratings do not differ between males and females in the population. 24.3 23 1.3
Now we need to see if they differ among the grades. Is the average performace rating the same for all grades? 42.6 40 2.6
(Assume variances are equal across the grades for this ANOVA.) Here are the data values sorted by grade level. 24.8 23 1.8
The rating values sorted by grade have been placed in columns I - N for you. A B C D E F 75.7 67 8.7
Null Hypothesis: Ho: means equal for all grades 90 80 100 90 85 70 64.4 57 7.4
Alt. Hypothesis: Ha: at least one mean is unequal 80 75 100 65 100 100 48 48 0
100 80 90 75 95 95 61.3 57 4.3
Anova: Single Factor 90 70 80 90 55 95 62.9 57 5.9
80 95 80 95 90 95 66.3 57 9.3
SUMMARY 85 80 95 95 60 57 3
Groups Count Sum Average Variance 65 90 90 67.2 57 10.2
A 15 1265 84.3333333333 153.0952380952 70 75
B 7 570 81.4285714286 72.619047619 95 95
C 5 450 90 100 60 90
D 5 415 83 157.5 90 95
E 12 1045 87.0833333333 152.0833333333 75 80
F 6 550 91.6666666667 116.6666666667 95
90
100
ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 519.2023809524 5 103.8404761905 0.7789853558 0.5702154774 2.4270401198
Within Groups 5865.2976190476 44 133.3022186147
Total 6384.5 49
Interpretation of test results:
What is the p-value: 0.57 If the ANVOA was done correctly, this is the p-value shown.
Is P-value < 0.05? No
Do we REJ or Not reject the null? Do not Reject null hypotheis
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: This means that performance ratings donot differ by grades
3 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?
Use the input table to the right to list salaries under each grade level.
(Assume equal variance, and use the analysis toolpak function ANOVA.)
Null Hypothesis: Ho: salaries equal for all grades If desired, place salaries per grade in these columns
Alt. Hypothesis: Ha: at least one salary is unequal A B C D E F
21.5 27.2 41 47.7 60.3 76
Anova: Single Factor 23.6 35 41 51.4 61 74.3
24.8 35 47.1 56.4 66.1 74.4
SUMMARY 22.1 34.2 42.5 48.8 69.6 75
Groups Count Sum Average Variance 24.6 27.4 42.6 48 60.6 79.9
A 15 355.8 23.72 1.2902857143 24.9 27.3 58.9 75.7
B 7 220.6 31.5142857143 15.6214285714 23.4 34.5 64.4
C 5 214.2 42.84 6.273 25.1 61.3
D 5 252.3 50.46 13.148 23 62.9
E 12 758.6 63.2166666667 11.7142424242 24.1 66.3
F 6 455.3 75.8833333333 4.3336666667 24.4 60
22.7 67.2
22.5
ANOVA 24.3
Source of Variation SS df MS F P-value F crit 24.8
Between Groups 17941.9336285714 5 3588.3867257143 464.3773123402 6.95561289654578E-37 2.4270401198
Within Groups 340.0015714286 44 7.7273084416
Total 18281.9352 49
Note: Sometimes we see a p-value in the format of 3.4E-5; this means move the decimal point left 5 places. In this example, the p-value is 0.000034
What is the p-value: 6.95E-37
Is P-value < 0.05? Yes
Do we REJ or Not reject the null? Reject Null hypothesis
If the null hypothesis was rejected, calculate the effect size value (eta squared): 0.9814023205
If calculated, what is the meaning of effect size measure: It means that the treatment effect quite large
Interpretation: 98% of total variation is acounted for by the treatment effect
4 The table and analysis below demonstrate a 2-way ANOVA with replication. Please interpret the results.
Note: These values are not the same as the data the assignment uses. The purpose of this question is to analyze the result of a 2-way ANOVA test rather than directly answer our equal pay question.
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 Ha: 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: 0.5389
Is P-value < 0.05? No
Do you reject or not reject the null hypothesis: Do not reject
If the null hypothesis was rejected, what is the effect size value (eta squared):
Meaning of effect size measure:
For Ho: Average compas are equal for all degrees Ha: Average compas are not equal for all grades
What is the p-value: 0.30883
Is P-value < 0.05? No
Do you reject or not reject the null hypothesis: Do not reject
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: 0.30189
Is P-value < 0.05? No
Do you reject or not reject the null hypothesis: Do not reject
If the null hypothesis was rejected, what is the effect size value (eta squared):
Meaning of effect size measure:
What do these three decisions mean in terms of our equal pay question: The decisions mean that average compas do not differ based on gender and degrees
5.   Using the results up thru this week, what are your conclusions about gender equal pay for equal work at this point?
Upto this point we see that pay differs by gender.