Bsb123 : Data Analysis : Assessment Answer

Answer:

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)

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.

(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.

 

Summary Report:

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.   

 


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