Statistics assignment Wk 2

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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
13	42	1.050	40	30	100	2	1	4.7	1	F	C
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			45	77	1.087	1.149
	Is this a 1 or 2 tail test?			Is this a 1 or 2 tail test?
	- why?			- why?
	P-value is:			P-value is:			45	55	1.052	1.145
	Is P-value > 0.05?			Is P-value > 0.05?			45	65	1.157	1.140
	Why do we not reject Ho?			Why do we not reject Ho?
Interpretation:
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:
	Ha:
	Test to use:
	Place B43 in Outcome range box.

	P-value is:
	Is P-value < 0.05?
	Reject or do not reject Ho:
If the null hypothesis was rejected, what is the effect size value:
	Meaning of effect size measure:
	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?
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:
	Ha:
	Statistical test to use:
	Place B75 in Outcome range box.
	What is the p-value:
	Is P-value < 0.05?
	Reject or do not reject Ho:
If the null hypothesis was rejected, what is the effect size value:
	Meaning of effect size measure:
	Interpretation:
4	Since performance is often a factor in pay levels, is the average Performance Rating the same for both genders?
	Ho:
	Ha:
	Test to use:
	Place B106 in Outcome range box.
		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:
		Meaning of effect size measure:
		Interpretation:
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?

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

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.

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

High

Males

Females

<Reminder: standard error is the sample standard deviation divided by the square root of the sample size.>

Interpretation:

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

High

Yes/No

Can the means be equal?

Why?

How does this compare to the week 2, question 2 result (2 sampe t-test)?

Why is using a two sample tool (t-test, confidence interval) a better choice than using 2 one-sample techniques when comparing two samples?

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

Total

Do manual calculations per cell here (if desired)

M Grad

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:

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)

OBS COUNT - m

OBS COUNT - 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

Create a correlation table for the variables in our data set. (Use analysis ToolPak or StatPlus:mac LE function Correlation.)

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):

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?

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?

Does this help us answer our equal pay for equal work question?

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

ANOVA

Significance F

Regression

17762.2996738743

2960.383278979

419.1516111294

1.8121523852609E-36

Residual

303.7003261257

7.062798282

Total

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?

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?

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?

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?