Statistics Analysis Report: Analysis of Employee Data Statistics

This report presents a statistical analysis of employee data, addressing several key questions posed by the Human Resources Manager, Lee Matson. The analysis includes calculations of the mean hours worked, estimation of the population mean, and tests of hypotheses regarding the mean work hours and proportion of employees likely to stay. A one-sample t-test revealed that the mean weekly work hours are less than 48 hours, and a Z-test indicated that more than 40% of employees are likely to stay. The report also examines the difference in mean work hours between males and females using a paired sample t-test, and calculates the required sample sizes for given confidence intervals. Furthermore, a simple linear regression was performed to assess the relationship between years of service and salary. The report concludes with recommendations regarding sample size and the impact of years worked on salary. The report provides valuable insights into employee work patterns, job satisfaction, and compensation structures, offering data-driven recommendations for management.

Statistics 1
Name
Tutor
Institution
Date

Statistics 2
PART 1: ANALYSIS
Question one
a. Mean hours worked
Estimate of population mean
Estimate of population mean=S . E+ 1
n ∑
i=1
n
xi
1
n ∑
i=1
n
xi= 20371
450 =45.27
S . E= δ
√ n = 9.81
√ 450 =0.46
Mean Estimate=45.27 ± 0.46
b. The proportion of employees who are “very likely to stay” in their jobs
Row Labels
Count of Stay
Org
Likely 190
not sure 22
Unlikely 117
Very likely 80
Very unlikely 41
Grand Total 450
Table 1
Sample Proportion , ῥ= very likely
total
Sample Proportion , ῥ= 80
450 =0.18
Estimate of the proportion , P=S . E ± ῥ
S . E= √ pq
n = √ 0.18 ×0.82
450 =0.02
Proportion Estimate=0.18 ±0.02

Statistics 3
Question two
Testing whether mean work hours are 48 hours a week.
To test the claim above, a one sample t-test is employed
Hypothesis
H0: The mean number of hours worked across the industry is 48 per week
Versus
H1: The mean number of hours worked across the industry is less than 48 per week
The test results were as shown below at 95% confidence interval
One-Sample Test
Test Value = 48
t df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the
Difference
Lower Upper
Work hours -5.905 449 .000 -2.73111 -3.6401 -1.8221
Table 2
The table above shows the results of a t-test. It can be seen that the p-value computed
which is 0.00 is less than the level of significance which is 0.05. This guides the test not
to accept the null hypothesis and not reject the alternative. The conclusion is therefore
that the mean number of hours worked across the industry is less than 48 per week.
Question 3
Test for population proportion

Statistics 4
Row Labels Count of StayOrg
Likely 190
not sure 22
Unlikely 117
Very likely 80
Very unlikely 41
Grand Total 450
Table 3
Sample Proportion , ῥ= likely
total
Sample Proportion , ῥ= 190
450 =0.42
Hypothesis
H0: Proportion of employees likely to stay in the job is 40%.
Versus
H1: Proportion of employees likely to stay in the job is more than 40%.
Z= √ pq
n = √ 0.4 ×0.6
450
Z=0.023
Z-computed is less than Z-tabulated at 95% (1.96). This indicates that the null
hypothesis is rejected and alternative accepted. It is therefore concluded that the
proportion of employees likely to stay in the job is more than 40%.
Question 4
Test for equality of mean between males and females
Hypothesis
H0: There is no difference in mean hours worked between males and females.
Versus
H1: There is a significant difference in the in hours worked weekly between males and
females.

Statistics 5
t-Test: Two-Sample Assuming Unequal Variances
female male
Mean 41.79473684 47.80769231
Variance 40.48145363 122.032373
Observations 190 260
Hypothesized Mean Difference 0
df 427
t Stat -7.278862674
P(T<=t) one-tail 8.14773E-13
t Critical one-tail 1.648429975
P(T<=t) two-tail 1.62955E-12
t Critical two-tail 1.965535168
Table 4
The table above shows the results of a paired sample t-test. It can be seen that the p-
value computed which is 0.00 is less than the level of significance which is 0.05. This
guides the test not to accept the null hypothesis and not reject the alternative. The
conclusion is therefore that there is a significant difference in the in hours worked
weekly between males and females.
Question 5
a. Finding sample size given interval limit
Limits = 2
Z-value = 1.645 (for 95% confidence interval)
Limits=Zvalue × δ
√n
2=1.645 × 9.8
√n
√n= 1.645 ×9.8
2
n= ( 8.06 )2
n=64.96 ≈ 65

Statistics 6
b. Finding sample size for a proportion given proportion limits
Limits = 0.05
Z-value = 1.645 (for 95% confidence interval)
Proportion = 0.18
limit =Zvalue × √ pq
n
0.05=1.645× √ 0.18 × 0.82
n
( 0.05
1.645 )
2
= 0.1476
n
( 0.030395 )2= 0.1476
n
n= 0.1476
0.00092=160.43
n=160
Question 6

Statistics 7
a. Simple linear regression result
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.298312
R Square 0.08899
Adjusted R Square 0.086952
Standard Error 13.45018
Observations 449
ANOVA
df SS MS F Significance F
Regression 1 7899.183 7899.183 43.66424 1.11E-10
Residual 447 80865.6 180.9074
Total 448 88764.78
CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%
Intercept 21.09627 1.420111 14.85536 7.37E-41 18.30535 23.88719
Work years 0.419439 0.063475 6.607892 1.11E-10 0.294691 0.544186
Table 5
b. Regression model
Salary last year=0.42 ( work years ) +21.1
c. The intercept (21.1) refers to the amount of salary in case the number of years is
zero. It is a constant amount regardless of the number of years worked. The
coefficient years (0.42) stands for the rate of change that occurs in salary incase
work years changes by one unit. The value of R-squared is 0.09. This means that
workers years in job can only explain 9% of the variation that occurs in salary.
d. Prediction of salary when work years is 39
Salary last year=0.42 ( w ork years ) +21.1
Salary last year=0.42 ( 39 ) +21.1
Salary=37.48
PART 2: RESPONSE TO LEE

Statistics 8
Dear Lee,
The following is the report you requested regarding further analysis of your data.
The analysis found that the weekly mean hours worked by the sample employees was
45.27 hours. This mean was just specific for the sample of 450 employees presented in
the data. However, this report also established where the population parameters lie in
terms of mean week hours by calculating the estimate of population mean. It found that
the mean week hours for the total population of workers would lie within 0.46 hours
above or below the sample mean (45.27 0.46). As for the proportion of employees
who attested that they are likely to stay in the organization, the sample proportion was
0.18. However, the estimate for the population proportion after calculation was found to
lay 0.02 units above or below the sample proportion.
In establishing whether the mean week a work hour was less than 48 hours, a one
sample t-test was employed. The results indicated that the alternative hypothesis, that
the mean week hours are less than 48 hours should be accepted. The report therefore
concludes that in deed the mean week hours are less than 48 hours.
The report also brings to the knowledge of management that there were no significant
differences in weekly work hours between the males and the females. This was arrived
at after a paired sample t-test for difference in means gave a p-value which was less
than the level of significance thereby making the report to reject the null hypothesis the
mean weekly work hours between the males and females are equal. The senior
management of your company hypothesized that more than 40% of the employees
were likely to stay in the organization. We conducted Z-test for the proportion. The
report was led to accept the alternative hypothesis and this confirmed that in deed the
proportion of those who are likely to stay in the organization is more than 40%. The
For attention: Lee Matson, Human Resources Manager Date…………..
From: John Frank, Senior Analyst.
Regarding: Further analysis of data

Statistics 9
actual calculation done from table 3 showed a proportion of 42% again confirming the
same.
It was also found that for the mean limits to be within a range of 2 and at 95%
confidence, the sample size should be 65 and not 450 for the estimates to be very
accurate. To add on, for the sample proportion to be within 0.05, the sample size should
be 160 to accurately estimate the proportion.
In order to determine whether and the extent to which number of years worked affected
the salaries, a simple linear regression was run. This method was appropriate at it
shows the linear relationship between dependent and independent variable. In this
case, the dependent variable was salary while the independent variable was the
number of years worked. The results showed that an employee with zero years will earn
a salary of 21.1 dollars. It was found that the number of years worked did not greatly
affect the salary earned since it was only able to explain 9% of the variation in salary as
indicated by the value of R-squared.