Boxplots, T-tests, And Regression Analysis On Academic Salaries

Male and Female Salaries Boxplots

Task 1 (Boxplots and t-test)

1 a)

Boxplot for salaries of male and female academics

                                                            

We can observe that mean salary of male is higher than female. Male salaries has more variations than the female salaries. Male salaries are more skewed than female salaries. We can observe that there are 3 outliers in the male salaries and one in female salary.

The following table shows the descriptive statistics for female and male salaries

                                                                                 

1 b)

Here we are interesting in testing the claim that male academics earns on an average more than female.

Here our null hypothesis is that there is no significant difference between average earning of male and female and alternative hypothesis is that  male academics earns on an average more than female.

We run independent two sample t test at 1%.

t-Test: Two-Sample Assuming Unequal Variances
	Male Salary	Female Salary
Mean	101136	81665.45
Variance	6.36E+08	2.25E+08
Observations	133	66
Hypothesized Mean Difference	0
df	191
t Stat	6.801746
P(T<=t) one-tail	6.46E-11
t Critical one-tail	2.34603
P(T<=t) two-tail	1.29E-10
t Critical two-tail	2.601814

Now our alternative hypothesis is one sided, so we compare t Critical one-tail with t stat. And here t Stat = 6.801746  > t Critical one-tail = 2.34603, so we reject null hypothesis. i.e. we support the claim that that male academics earns on an average more than female.

2 a)

Here we are interesting in testing the claim that male assistant professor earns on an average more than female assistant professor.

Here our null hypothesis is that there is no significant difference between average earning of male assistant professor and female assistant professor and alternative hypothesis is that male assistant professor earns on an average more than female assistant professor.

We run independent two sample t test at 1%.

t-Test: Two-Sample Assuming Unequal Variances
	Male Salary	Female Salary
Mean	80117.43	76903.62
Variance	2.52E+08	1.94E+08
Observations	35	42
Hypothesized Mean Difference	0
df	68
t Stat	0.935136
P(T<=t) one-tail	0.176514
t Critical one-tail	2.382446
P(T<=t) two-tail	0.353027
t Critical two-tail	2.650081

Now our alternative hypothesis is one sided, so we compare t Critical one-tail with t stat. And here t Stat = 0.935135  < t Critical one-tail = 2.382446, so we fail to reject null hypothesis.

2 b)

Here we are interesting in testing the claim that male associate professor earns on an average more than female associate professor.

Here our null hypothesis is that there is no significant difference between average earning of male associate professor and female associate professor and alternative hypothesis is that male associate professor earns on an average more than female associate professor.

We run independent two sample t test at 1%.

 

t-Test: Two-Sample Assuming Unequal Variances
	Male Salary	Female Salary
Mean	96503.53	86039.5
Variance	2.7E+08	1.15E+08
Observations	47	20
Hypothesized Mean Difference	0
df	54
t Stat	3.086352
P(T<=t) one-tail	0.001597
t Critical one-tail	2.39741
P(T<=t) two-tail	0.003194
t Critical two-tail	2.669985

Now our alternative hypothesis is one sided, so we compare t Critical one-tail with t stat. And here t Stat = 3.086352  > t Critical one-tail = 2.39741, so we reject null hypothesis. i.e. we support the claim that male associate professor earns on an average more than female associate professor.

2 c)

Test for Differences in Male and Female Earnings

Here we are interesting in testing the claim that male professor earns on an average more than female professor.

Here our null hypothesis is that there is no significant difference between average earning of male professor and female professor and alternative hypothesis is that male professor earns on an average more than female professor.

We run independent two sample t test at 1%.

t-Test: Two-Sample Assuming Unequal Variances
	Male Salary	Female Salary
Mean	119829.6	109794.5
Variance	5.73E+08	6962348
Observations	51	4
Hypothesized Mean Difference	0
df	48
t Stat	2.786449
P(T<=t) one-tail	0.003805
t Critical one-tail	2.406581
P(T<=t) two-tail	0.00761
t Critical two-tail	2.682204

3).

Now our alternative hypothesis is one sided, so we compare t Critical one-tail with t stat. And here t Stat = 2.786449  > t Critical one-tail = 2.406581, so we reject null hypothesis. i.e. we support the claim that male professor earns on an average more than associate professor.

The following table shows the correlation matrix between the Salary, Age and Years of service.

	Salary	Age	Years of Service
Salary	1	0.407	0.427
Age	0.407	1	0.942
Years of Service	0.427	0.942	1

We can say that there is moderate positive correlation between salary and age, salary and year of service. We observed that there is strong positive correlation between age and service of years.

Bonus Question:

Following table shows the gender gap in mean salary.

	Count		Mean Salary
	Female	Male	Female	Male	Gender Gap
Assistant Professor	42	35	76903.62	80117.43	4.0%
Associate Professor	20	47	86039.5	96503.53	10.8%
Professor	4	51	109794.5	119829.6	8.4%
Total	66	133	81665.45	101136	19.3%

We can see that total gender gap is 19.3% whereas gender gap rank wise is 4%, 10.8% and 8.4% for assistant professor, associate professor and professor respectively.

Yes there is association with simpson’s paradox as we can see that gender gap as a whole is more than rank wise. It is mainly due to the number of female academics rank wise.

4).

Step 1: Gender Only

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.38084
R Square	0.145039
Adjusted R Square	0.140699
Standard Error	22369
Observations	199
ANOVA
	df	SS	MS	F	Significance F
Regression	1	1.67E+10	1.67E+10	33.41988	2.87E-08
Residual	197	9.86E+10	5E+08
Total	198	1.15E+11
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	81665.45	2753.434	29.6595	1.39E-74	76235.47	87095.44	76235.47	87095.44
Male	19470.53	3368.025	5.780993	2.87E-08	12828.52	26112.54	12828.52	26112.54

Regression Equation:

Salary = 81665.45 + 1947053 × (Male)

(Male) = 1 is respondent is male other wise 0.

Step 2: Gender and School

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.54559
R Square	0.297668
Adjusted R Square	0.283187
Standard Error	20430.4
Observations	199
ANOVA
	df	SS	MS	F	Significance F
Regression	4	3.43E+10	8.58E+09	20.55566	3.89E-14
Residual	194	8.1E+10	4.17E+08
Total	198	1.15E+11
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	82796.28	4089.603	20.24556	1.01E-49	74730.49	90862.07	74730.49	90862.07
Male	17582.4	3109.693	5.654064	5.54E-08	11449.26	23715.55	11449.26	23715.55
BUSINESS	23340.81	5412.916	4.312059	2.57E-05	12665.09	34016.53	12665.09	34016.53
LIBERAL STUDIES	-2637.14	4365.554	-0.60408	0.546497	-11247.2	5972.899	-11247.2	5972.899
SCIENCES	-6266.5	4471.514	-1.40143	0.162684	-15085.5	2552.525	-15085.5	2552.525

Regression Equation:

Salary = 88706.14 + 17582.4 × (Male) + 23340.81 × (Business) – 2637.14 × (Liberal Studies) – 6266.5 × (Science)

(Male) = 1 is respondent is male other wise 0.

(Business) = 1 if respondent from business school otherwise 0

(Liberal Studies) = 1 if respondent from Liberal Studies school otherwise 0

(Science) = 1 if respondent from Science school otherwise 0

Step 3: Gender, School and Rank

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.816276
R Square	0.666306
Adjusted R Square	0.655878
Standard Error	14155.66
Observations	199
ANOVA
	df	SS	MS	F	Significance F
Regression	6	7.68E+10	1.28E+10	63.89621	3.72E-43
Residual	192	3.85E+10	2E+08
Total	198	1.15E+11
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	91061.45	3330.746	27.33965	4.07E-68	84491.9	97631.01	84491.9	97631.01
Male	4294.581	2348.292	1.828811	0.068979	-337.181	8926.344	-337.181	8926.344
BUSINESS	24521.53	3758.249	6.524721	5.9E-10	17108.77	31934.28	17108.77	31934.28
LIBERAL STUDIES	-7201.49	3051.139	-2.36026	0.019265	-13219.5	-1183.43	-13219.5	-1183.43
SCIENCES	-5894.05	3098.857	-1.90201	0.058667	-12006.2	218.1287	-12006.2	218.1287
Assistant Professor	-13513.7	2473.472	-5.46346	1.44E-07	-18392.4	-8635.05	-18392.4	-8635.05
Professor	26523.75	2638.358	10.05313	2.27E-19	21319.86	31727.64	21319.86	31727.64

Regression Equation:

Salary = 91061.45 + 4294.581 × (Male) + 24521.53 × (Business) – 7201.49 × (Liberal Studies) – 5894.05 × (Science) – 13513.7 × (Assistant Professor) + 26523.75 × (Professor)

Where

(Male) = 1 is respondent is male other wise 0.

(Business) = 1 if respondent from business school otherwise 0

(Liberal Studies) = 1 if respondent from Liberal Studies school otherwise 0

(Science) = 1 if respondent from Science school otherwise 0

(Assistant Professor) = 1 if respondent is assistant professor otherwise 0

(Professor) = 1 if respondent is professor otherwise 0

Step 4: Gender, School, Rank and Years of Service

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.836086
R Square	0.699039
Adjusted R Square	0.688009
Standard Error	13478.6
Observations	199
ANOVA
	df	SS	MS	F	Significance F
Regression	7	8.06E+10	1.15E+10	63.37627	1.83E-46
Residual	191	3.47E+10	1.82E+08
Total	198	1.15E+11
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	83501.02	3579.05	23.3305	8.24E-58	76441.48	90560.56	76441.48	90560.56
Male	3450.289	2243.634	1.537813	0.125749	-975.194	7875.773	-975.194	7875.773
BUSINESS	26906.36	3616.545	7.439797	3.33E-12	19772.86	34039.86	19772.86	34039.86
LIBERAL STUDIES	-8068.44	2911.425	-2.7713	0.006135	-13811.1	-2325.77	-13811.1	-2325.77
SCIENCES	-6177.09	2951.294	-2.09301	0.037669	-11998.4	-355.778	-11998.4	-355.778
Assistant Professor	-10169.2	2466.833	-4.12238	5.59E-05	-15035	-5303.49	-15035	-5303.49
Professor	23551.59	2595.423	9.074277	1.41E-16	18432.21	28670.96	18432.21	28670.96
Years of Service	593.5535	130.2279	4.557805	9.21E-06	336.6839	850.4232	336.6839	850.4232

Regression Equation:

Salary = 83501.02 + 3450.289 × (Male) + 26906.36 × (Business) – 8068.44 × (Liberal Studies) – 6177.09 × (Science) – 10169.2 × (Assistant Professor) + 23551.59 × (Professor) +593.5535 × Years of Service

Where

(Male) = 1 is respondent is male other wise 0.

Test for Differences in Male and Female Assistant Professors Earnings

(Business) = 1 if respondent from business school otherwise 0

(Liberal Studies) = 1 if respondent from Liberal Studies school otherwise 0

(Science) = 1 if respondent from Science school otherwise 0

(Assistant Professor) = 1 if respondent is assistant professor otherwise 0

(Professor) = 1 if respondent is professor otherwise 0

Step 5: Gender, School, Rank, Years of Service and Age

 

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.836138
R Square	0.699127
Adjusted R Square	0.686459
Standard Error	13512.05
Observations	199
ANOVA
	df	SS	MS	F	Significance F
Regression	8	8.06E+10	1.01E+10	55.18698	1.47E-45
Residual	190	3.47E+10	1.83E+08
Total	198	1.15E+11
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	81256.74	10173.08	7.987424	1.29E-13	61190.05	101323.4	61190.05	101323.4
Male	3538.509	2280.116	1.551898	0.122351	-959.085	8036.102	-959.085	8036.102
BUSINESS	26979.21	3638.663	7.414593	3.92E-12	19801.84	34156.58	19801.84	34156.58
LIBERAL STUDIES	-8059.37	2918.904	-2.76109	0.006326	-13817	-2301.75	-13817	-2301.75
SCIENCES	-6197.97	2959.943	-2.09395	0.037591	-12036.5	-359.404	-12036.5	-359.404
Assistant Professor	-10153.4	2473.866	-4.10426	6.02E-05	-15033.2	-5273.63	-15033.2	-5273.63
Professor	23431.59	2651.176	8.83819	6.54E-16	18202.08	28661.11	18202.08	28661.11
Years of Service	521.9534	330.5714	1.578943	0.116012	-130.108	1174.015	-130.108	1174.015
Age	72.63425	308.0866	0.235759	0.813873	-535.075	680.3438	-535.075	680.3438

Regression Equation:

Salary = 81256.74 + 3538.509 × (Male) + 26979.21 × (Business) – 8059.37 × (Liberal Studies) – 6197.97 × (Science) – 10153.4 × (Assistant Professor) + 23431.59 × (Professor) + 521.9534 × Years of Service + 72.63425 × Age

Where

(Male) = 1 is respondent is male other wise 0.

(Business) = 1 if respondent from business school otherwise 0

(Liberal Studies) = 1 if respondent from Liberal Studies school otherwise 0

(Science) = 1 if respondent from Science school otherwise 0

(Assistant Professor) = 1 if respondent is assistant professor otherwise 0

(Professor) = 1 if respondent is professor otherwise 0

Significance of Coefficient:

Here we test the significance of coefficient, it is two sided hypothesis. Above table show the P-value of significance test. IF P-Value < 0.05, we claim that variable is significant otherwise not.

So,

Business School, Liberal school, Assistant professor and Professor are significant variables.

5).

1. a) R square is used to measure the adequacy of the model.

Step	R Square
1	0.145039
2	0.297668
3	0.666306
4	0.699039
5	0.699127

Model fitted in Step 5 is more adequate . But there is very little increment in step 5 from step 4.

1. b) Coefficient and their significance in step 5:

	Coefficients	Standard Error	t Stat	P-value
Intercept	81256.74	10173.08	7.987424	1.29E-13
Male	3538.509	2280.116	1.551898	0.122351
BUSINESS	26979.21	3638.663	7.414593	3.92E-12
LIBERAL STUDIES	-8059.37	2918.904	-2.76109	0.006326
SCIENCES	-6197.97	2959.943	-2.09395	0.037591
Assistant Professor	-10153.4	2473.866	-4.10426	6.02E-05
Professor	23431.59	2651.176	8.83819	6.54E-16
Years of Service	521.9534	330.5714	1.578943	0.116012
Age	72.63425	308.0866	0.235759	0.813873

 (Male) = 1 is respondent is male other wise 0.

(Business) = 1 if respondent from business school otherwise 0

(Liberal Studies) = 1 if respondent from Liberal Studies school otherwise 0

(Science) = 1 if respondent from Science school otherwise 0

(Assistant Professor) = 1 if respondent is assistant professor otherwise 0

(Professor) = 1 if respondent is professor otherwise 0

If respondent is male then there is $3538.509 increment in salary.

If respondent is from business school then there is $26979.21 increment in salary.

If respondent is from liberal study school then there is $8059.37 decrement in salary.

If respondent is from science school then there is $6197.97 decrement in salary.

If respondent is assistant professor then there is $10153.4 decrement in salary.

If respondent is professor then there is $23431.59 increment in salary.

If year of service increased by 1 year salary increased by $521.9534 and if age is increases by 1 year then salary is increased by 72.63425

Significance of Coefficient:

Here we test the significance of coefficient, it is two sided hypothesis. Above table show the P-value of significance test. IF P-Value < 0.05, we claim that variable is significant otherwise not.

So,

Business School, Liberal school, Assistant professor and Professor are significant variables.

6).

From correlation analysis, salary is significantly related with age but in step 5, we came across the conclusion that age is not significant factor for salary.

We have data of 199 academics from the particular college. We noted the school in which they work, their rank, gender, age, year of service and salary. There are 66 female and 133 male academics in the data.  We observed that the mean salary of male academics is more than female academics. We also noted that there is more variation in the male salary than female salary.

We used two sample t-test for comparing mean salary of male and female. We observed that male academics have more pay than female academics. We also compare mean salary of male and female for assistant professor, associate professor and professor. We observed that there is no significant difference between male assistant professor and female assistant professor whereas male associate professor and professor earns more salary than female associate professor and professor.

From the correlation analysis of salary, age and years of service. We observed that there is positive and significant relationship between this variables.

We used stepwise regression to the salary amount we add one by one variable gender, school, rank, years of service and age. We used female, health and associate professor as a reference variable for nominal variables. We observed that R² is increased from 0.17 to 0.7. In step 1, we used gender as predictor variable and it is found to be significant. In step 2, we add school variable, in third we and rank and so on. The regression equation for the step 5 is as follows:

Salary = 81256.74 + 3538.509 × (Male) + 26979.21 × (Business) – 8059.37 × (Liberal Studies) – 6197.97 × (Science) – 10153.4 × (Assistant Professor) + 23431.59 × (Professor) + 521.9534 × Years of Service + 72.63425 × Age

where

(Male) = 1 is respondent is male other wise 0.

(Business) = 1 if respondent from business school otherwise 0

(Liberal Studies) = 1 if respondent from Liberal Studies school otherwise 0

(Science) = 1 if respondent from Science school otherwise 0

(Assistant Professor) = 1 if respondent is assistant professor otherwise 0

(Professor) = 1 if respondent is professor otherwise 0

In step 5, we found that school and rank are only the significant variables for predicting the salary of the employee. Gender, year of service and age are not significant factor for predicting the salary of the employee.