Sta 2023 : Statistics : Assessment Answers

1. Before you begin your analysis you are required to take a random sample of size 100from the 171 cases in the data file. Use the Data and Generator.xls to do this. Open this file and go to the first worksheet labelled Macro. Click on Generate and enter the number of rows of data required in the random sample. If you have repeated cases your final sample size may be less than 100. This will not be a problem. Answers to the questions below are to be based on this random sample of up to 100 cases. Make sure you keep a safe copy of your sample since you cannot use the Random Sample Generator to reproduce the first sample.

2. Data List:Provide a printout of the data in your sample, with ID numbers in ascending order.

3. Introduction and Variable List: Provide a summary table listing each variable with informationunder the following column headings: Variable Name, Variable Description (see the information above) and Variable Type (qualitative or quantitative). If the variable is qualitative state whether it is nominal or ordinal; if it is quantitative you need to state whether it is discrete or continuous.

4. Descriptive statistics: Historically the company believed that average salary of male is greaterthan the average salary of female. Determine the mean, median, and standard deviation of current salary data for male and female. Explain the results in the context of the company’s believe. (10 marks)

5. Presentation of data: Draw Histogram and Boxplots for both male and female for current salary Explain the graphs. (10 marks)

PART 2:

6.Hypothesis testing: Use the above information to carry out a t-test to see the averagecurrent salary for male is significantly greater than the average current salary for female.

Explain the result in the light of the question.

7.Comparing gender and positions:

Prepare a Pivot table that shows average current salary for males and females according to their Position within the company. Think carefully about the layout of rows and columns of your table. Place gender in row, positions in column, and current salaries in value box. Then summarise values as average. Explain the results.

8.Comparing Departments: The head of Human Resources wants to compare the structure of thefour departments by gender within the Company.

a.Draw a Pivot table with gender in row, department in column and gender in value.
b.Determine the joint probability for gender and department. Explain the result.
c.Determine the marginal probability and explain the result.
d.Determine the conditional probability by column and explain the result.
e.Determine the conditional probability by row and explain the result.

PART 3:

9.Average salary increase per year: There have been complaints in this company that femaleemployees have been given lower salary increases than their male colleagues.

Examine the issue: Have the salary increases for females been lower?

a). Create a new column, called Length Emplwhich shows the length of time a person has been employed in the company, as of the end of December 2014. Assume that if an employee started in 2008 than they have been employed for 6 years; if they started in 1997 than they have been employed for 17 years, and so on. Create a second new column, called Avg Incr which shows the average increase in salary for each employee since the year they started work for the company.

State the formulae you have used to produce these new columns. Print out the ID column and the new data – 3 columns. This should be in ascending order of the ID numbers.

b). Use Avg Incrto create a histogram showing the distribution of the average salary increase per year. This histogram has been created to get some idea of the distribution of the data. Comment upon the shape of this distribution: Is it symmetric? Is it skewed? If so which way? (5 marks)

c). Now provide side by side boxplots for Avg Incr, split on gender. Again, comment on what you see. Most importantly, provide your answer to the question: “Have the salary increases for females beenlower than those for males?” (10 marks)

d). Run regression analysis with Start Salaryand Current Salary for both male and female separately. Check carefully which variable will be dependent variable and which variable will be independent variable Again comment on what you see. Most importantly, provide your answer to the question:

Have the salary increases for females been lower than those for males?” Comment whether the regression analyses support or oppose the conclusion in (c).

10.Draw a suitable conclusion based on your data analysis.

Answer:

Part 1

A random data is derived through the macro developed for the generation of dataset. The dataset generated is saved in the excel sheet.

The sample data of 100 is presented in Appendix

Table 1: Summary Table of Variables

Variable Name	Variable Description	Variable Type
Gender	Prevalence of gender in the business	Qualitative - Nominal
Start Year	The year in which the employees started the job.	Quantitative - Discrete
Department	The department of the Employee.	Qualitative - Ordinal
Start Salary	The starting salary of the employees	Quantitative - Continuous
Current Salary	The present salary of the employees	Quantitative - Continuous
Years of Experience	The number of years of experience	Qualitative - Ordinal
Position	Based on the quality of work the current position of the employees.	Qualitative - Ordinal

Table 2: Summary Statistics for Current salary

	Male	Female
Mean	63180.79	64307.69
Median	60960	58920
Mode	63120	57120
Minimum	52080	42240
Maximum	119508	90600
Range	67428	48360
Variance	114606565.8	221572644.5
Standard Deviation	10705.45	14885.32
Coeff. of Variation	16.94	23.15
Skewness	3.21	0.47
Kurtosis	13.11	-1.11
Count	61	39
Standard Error	1370.69	2383.56

From the given sample it is found that the average salary of males and females is 63180.79 and 64307.69 respectively. Thus the belief of the company that the average salary of males is greater than females is found to be untrue for the current salary of the employees.

The deviation in males and females is 10705.45 and 14885.32 respectively. Thus, it can be seen that the variation in current salary for males is lower than females.

The median salary of males is 60960 while for females is 58920. Thus, it found that 50% of males gets below and above 60960. Similarly, 50% of the females get below and above 58920.

5).

Figure 1: Boxplot of Current Salary of Males and Females

From the above boxplot the distribution of current salary of Males and can be compared. It is found that the minimum salary of males is higher than females. Moreover, the maximum current salary of males is also higher than maximum current salary of females. The median current salary of males is slightly higher than females. It is also found that the current salary for both males and females is skewed right.

Figure 2: Histogram of Current Salary of Males

From the histogram it can be seen that the distribution of the current salary of males is skewed right. Thus, the mean of the current salary is higher than the median salary.

Figure 3: Histogram of Current Salary of Females

From the histogram it can be seen that the distribution of the current salary of females is skewed right. Thus, the mean of the current salary is higher than the median salary.

Part 2

6).

In order to analyse whether the average current salary of males is higher than females

Null Hypothesis: The average current salary of males and females are equal

Alternate Hypothesis: The average current salary of males is higher than females.

Table 3: Summary Statistics

Separate-Variances t Test for the Difference Between Two Means
(assumes unequal population variances)
Data
Hypothesized Difference	0
Level of Significance	0.05
Population 1 Sample
Sample Size	61
Sample Mean	63180.78689
Sample Standard Deviation	10705.4456
Population 2 Sample
Sample Size	39
Sample Mean	64307.69231
Sample Standard Deviation	14885.3164

Table 4: Independent Sample t-test

Upper-Tail Test
Upper Critical Value	1.6698
p-Value	0.6583
Do not reject the null hypothesis
Separate-Variance t Test Statistic	-0.4098

The average current salary of males is 63180.8 The average current salary of females is 64307.7.

From the independent sample t-test it is found that t(62) =-0.4098, p-value = 0.6583 at a = 0.05, level of significance.

The p-value is more than the level of significance. Hence we do not reject the Null hypothesis. Thus, the hypothesis test demonstrates that the average current salary of males and females are equal.

7).

Table 5: Relation of Average Current Salary to Gender and Position

Average of Current Salary	Position
Gender	1	2	3	Grand Total
Female	85080	85526	56036	64308
Male	97556	87660	60465	63181
Grand Total	90427	86000	58989	63620

From the above table it is found that the average current salary varies across gender and positions. For Males the average current salary is highest for position 1 followed by position 2 and the least is position 3. For females the average current salary is highest for Position 1, followed by Position 2 and the least for position 3.

Further, the average current salary for males is higher than females for all three positions.

8).

Part a

Table 6: Count of Gender across Departments

Count of Gender	Department
Gender	1	2	3	4	Grand Total
Females	6	8	15	10	39
Males	9	27	10	15	61
Grand Total	15	35	25	25	100

From the above table it is found that for departments 1,2 and 4 the number of males is higher than females. In department 3 the number of females is higher than the number of males.

The sample dataset has the highest number of employees (35) from Department 2. Further, in the sample dataset the number of employees (25) from department 3 and 4 are equal.

Part b

	Department
Gender	1	2	3	4	Total
Females	0.06	0.08	0.15	0.1	0.39
Males	0.09	0.27	0.1	0.15	0.61
Total	0.15	0.35	0.25	0.25	1

The joint probability provides information on proportion of gender in a department.

Thus the proportion of females in department 1 is 0.06. From the joint probability table, it is found that the lowest proportion of employees is females in department 1 – 0.06. The highest proportion of employees – males in department 2 is 0.27

Part c

	Department
Gender	1	2	3	4	Total
Females	0.06	0.08	0.15	0.1	0.39
Males	0.09	0.27	0.1	0.15	0.61
Total	0.15	0.35	0.25	0.25	1

The marginal probabilities show that the proportion of females is to males is 0.39 to 0.69. Further, the proportion of employees in department 1, 2,3,4 is 0.15:0.35:0.25:0.25.

Part d

Count of Gender	Department
Gender	1	2	3	4	Grand Total
Females	40.00%	22.86%	60.00%	40.00%	39.00%
Males	60.00%	77.14%	40.00%	60.00%	61.00%
Grand Total	100.00%	100.00%	100.00%	100.00%	100.00%

The above table presents the conditional probability by column. The proportion of female employees in the organization is 39.0%. Similarly, the proportion of male employees in the organization is 61.0%.

Part e

Count of Gender	Department
Gender	1	2	3	4	Grand Total
Females	15.38%	20.51%	38.46%	25.64%	100.00%
Males	14.75%	44.26%	16.39%	24.59%	100.00%
Grand Total	15.00%	35.00%	25.00%	25.00%	100.00%

The above table presents the conditional probability by row. The proportion of employees in department 2 is the highest at 35.0%. The proportion of employees in department 3 and 4 are similar at 25.0%. The proportion of employees is department 1 is the lowest at 15.0%

Part 3

9).

Part a

A new variable “Length Empl” is created. To create the new column is MS-EXCEL we use the command:

=2014-C2

Here 2014 refers to the time when the management wants to know the duration of employment.

A second variable “Avg Incr” is created. The command used to create the variable:

=F2-E2

Where “F2” refers to the current salary of the employees and “E2” the starting salary of the employees.

The dataset is presented in Appendix.

Part b

Figure 4: Distribution of average Income

The distribution of average increase in Salary is left skewed. Thus the mean salary increase is greater than the median salary increase.

Part c

Figure 5: distribution of average increase in Salary based on gender

From the boxplot it is found that the average salary increase for both males and females is normally distributed. However, it is found that the minimum increase in Salary of females is higher than males. On the other hand, the maximum salary increase is more for males than females.

Thus for maximum salary increases the males have had a higher salary increase.

However, on the minimum scale, females have had a higher salary increase than males.

Part d

Females Salary

Calculations
b1, b0 Coefficients	0.8889	27641.4558
b1, b0 Standard Error	0.0585	2465.0302
R Square, Standard Error	0.7019	6828.0182
F, Residual df	230.7379	98.0000
Regression SS, Residual SS	10757426081.21	4568939654.95

Regression Statistics
Multiple R	0.8378
R Square	0.7019
Adjusted R Square	0.6988
Standard Error	6828.0182
Observations	100

Regression analysis is used to evaluate the current salary of females based on their starting salary. The starting salary is taken as the independent variable and the current salary as the dependent variable.

From the regression analysis the current salary of females can be predicted as:

Current Salary = 27641.5+0.8889*Starting salary

Further 70.19% of the variations in current salary of females can be predicted from their starting salary.

Males Salary

Calculations
b1, b0 Coefficients	0.8799	27952.7412
b1, b0 Standard Error	0.0599	2505.9337
R Square, Standard Error	0.6900	6843.6681
F, Residual df	215.9030	97.0000
Regression SS, Residual SS	10111986723.24	4543071979.31

Regression Statistics
Multiple R	0.8307
R Square	0.6900
Adjusted R Square	0.6868
Standard Error	6843.6681
Observations	99

Regression analysis is used to evaluate the current salary of males based on their starting salary.

From the regression analysis the current salary of males can be predicted as:

Current Salary = 27952.7+0.8799*Starting salary

Further 69.00% of the variations in current salary of males can be predicted from their starting salary.

Comparing the regression equation, it is seen that the basic salary of males is higher than females. Moreover, for one-unit increase in starting salary the increase in salary of females (0.8889) is higher than for males (0.8799).

Thus it is found that the salary increases for females is higher than males with the same starting salary. However, since the base salary of males is higher (27952.7) as compared to females (27641.5), thus the current salary for males would be higher.

Conclusion:

The present solution analysis the salaries of the employees of the organization. The initial analysis shows that the average salary of females is higher than males. However, the median salary of males is higher than females. Moreover, it is found that the average current salary of bot males and females is right skewed.

Even though the average current salary of females is higher than males, but the hypothesis tests that the average salary are equal.

It is found that the average current salary of both males and females varies across positions. In addition, there is variation in proportion of males and females across departments.

The investigation shows that the average increment in salary is left skewed. The average increment for genders is normally distributed. Further, it is seen that the minimum average increment for females is more than males. It is also found that the maximum average increment for males is higher than females.

The current salary on the basis of the starting salary is higher for males than females.

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