XYZ University Statistics Report: Graduate Data Analysis

This report presents a statistical analysis of graduate employment and salary data collected from a survey of 648 graduates from XYZ University. The analysis employs inferential statistical techniques, including ANOVA and hypothesis testing, to address key questions regarding employment status and salary levels across four different graduate disciplines: health, commerce, law, and engineering. The report investigates whether employment status varies significantly among the disciplines, compares salary levels, and examines income differences between commerce and health science graduates. Furthermore, it compares the employment proportions of law and engineering graduates. The findings, including the rejection or acceptance of null hypotheses, offer valuable insights for the university's career center, aiding in understanding employment trends and guiding future students in their career choices. The report concludes that while employment status may not differ significantly across disciplines, salary levels do, and the employment proportions of law and engineering graduates also show a statistically significant difference.

STATISTICS
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Introduction
XYZ University has send questionnaire to 100 graduates from four disciplines which pertains
to questions in relation to their employment status and salary level. However, only 648
graduates have responded to the survey. Using the available sample data along with suitable
inferential techniques (hypothesis testing), the objective of this report is to test the various
claims that have been presented. The given report would be submitted to the Director of the
career centre at the University and perhaps would help the University and the future students
in order to make prudent career choices based on their underlying objective.
Statistical Analysis
As highlighted in the previous section, there are certain questions which need to be answered
based on the given sample responses that have been collected from past students. It is
apparent that the objective of the given statistical analysis is to opine on the population trends
and the same has to be based on the sample information made available. As a result, the
relevant statistical techniques would be those that are fall under inferential statistics. The key
technique that would be deployed here is the testing of hypothesis using suitable tests based
on the underlying situation and satisfaction of corresponding assumptions.
Employment Status Comparison
It needs to be answered whether the employment status across the four group of graduates is
statistically different or not. It is apparent that the employment status is essentially categorical
data which is represented in the form of numerical values. The appropriate test deployed for
the same would be ANOVA single factor as the underlying means of the four graduate
disciplines need to be compared to determine if there is any significant difference between
these or not. An alternative in the form of T test cannot possibly be used here as the number
of variables whose average need to be compared are only four but a T test could only test the
same for two variables at a time. In total there are 648 samples which are distributed across
the four disciplines. Also, the samples in the various disciplines would be independent
samples as the underlying disciplines, interests and intellectual capabilities of the graduates
would not be dependent on each other.
For the ANOVA test, the following assumptions ought to be satisfied.

 The group sample needs to be drawn from a population that is normal.
 The underlying populations tend to have a common variance.
 The samples drawn are independent.
With regards to the population being sample, it is apparent that the questionnaire was sent to
1000 graduates but the method of selection of these graduates has not been outlined. It can be
either assumed that this constitutes the total graduates which passed in a particular year or the
sample of these 1000 was randomly drawn. Thus, assuming the above, the population can be
considered as normal. The samples drawn seem to be independent as 648 graduates
responded to the survey and based on the given information, it can be assumed that neither of
these was contacted by the university personally. Besides, in terms of population variance,
the sample variances are different and if the F test is used to compare the variances of each of
the two sample (according to discipline), the variance of the population would not come to be
same. However, despite the above, the ANOVA test is being applied in the given case.
Null Hypothesis (H0): μhealth= μCommerce= μLaw= μEngineering i.e. the employment status across the
various graduate discipline does not alter in a statistical significant manner.
Alternative Hypothesis (H1): Atleast one of the graduate disciplines would have an
employment status that is different from others in a statistical significant manner.
From the excel, it is apparent that SST = 1.1985
Also, from the excel SSE = 241.76
It is known that MST = SST
Degreeof FreedomTreatment
= 1.1985
4−1 =0.3995
Also, it is known that MSE = SSE
Degreeof FreedomError
¿ 241.76
648−4 =0.3754
F statistic= MST
MSE =( 0.3995
0.3754 )=1.064
The above value needs to be compared with the critical value of F.
Assuming a level of significance of 5% and df1= (4-1) = 3 and df2= (648-4) = 644, using the
F table, the critical value comes out as 2.619

It is apparent that the critical value of F is higher than the computed value of F statistic which
implies that the null hypothesis would not be rejected based on the available evidence. As a
result, the alternative hypothesis would not be accepted. Hence, it can be concluded that the
employment status of the different graduates can be assumed to be same.
Salary Comparison
Similar to the hypothesis test in the above case, here instead of the employment status, the
salary of the graduates across the four disciplines need to be compared. Considering that the
number of means that need to be compared is more than 2, hence ANOVA would be the best
alternative to perform the hypothesis test. The various assumptions of the ANOVA test along
with the relevant fulfilment by the given data has been already highlighted in the previous
hypothesis test and the same does not merit a discussion once again and thus, the hypothesis
testing must be carried out in the manner indicated below.
Null Hypothesis (H0): μhealth= μCommerce= μLaw= μEngineering i.e. the salary status across the various
graduate discipline does not alter in a statistical significant manner.
Alternative Hypothesis (H1): Atleast one of the graduate disciplines would have a salary level
that is different from others in a statistical significant manner.
From the excel, it is apparent that SST = 40961516882.59
Also, from the excel SSE = 43389225367.26
It is known that MST = SST
Degreeof FreedomTreatment
= 40961516882.59
4−1 =13653838960.86
Also, it is known that MSE = SSE
Degreeof FreedomError
= 43389225367.26
374−4 =117268176.67
F statistic= MST
MSE =( 13653838960.86
117268176.67 )=116.43
The above value needs to be compared with the critical value of F.
Assuming a level of significance of 5% and df1= (4-1) = 3 and df2= (374-4) = 370, using the
F table, the critical value comes out as 2.629.

It is apparent that the critical value of F is lower than the computed value of F statistic which
implies that the null hypothesis would be rejected based on the available evidence. As a
result, the alternative hypothesis would be accepted. Hence, it can be concluded that the
salary levels of the different graduates does tend to differ in a significant manner and
therefore cannot be assumed to be same.
Income comparison (Commerce Graduates v Health Science Graduates)
It is apparent that the given data is quantitative and the population averages for the two
independent samples need to be compared using the available sample data. For the given two
independent samples, the respective sample size for each exceeds 30, hence in line with the
Central Limit Theorem, it can be assumed that the given sample is normally distributed. As a
result, the use of z statistic for comparison of means may be considered. However, this would
not happen since the standard deviation of the respective population is not known and hence
it would be preferable to use T statistics ahead of the Z statistics. It is noteworthy that equal
variances would be assumed in the given case since the sample standard deviations of the two
samples do not differ significantly and also the sample sizes are similar.
The requisite assumptions for the deployment of the above mentioned test are highlighted
below.
 The underlying population from which the two samples have been drawn shown be
normal.
 The population standard deviation for the two samples must be equal.
 The sample must be independent.
Based on the given methodology of data collection from a group of 1000 graduates which are
taken to be randomly selected, it would be a fair assumption that the underlying population
would be normally distributed considering the respective size in each case and also jointly
would exceed 30. Further, considering the passive participation from the university along
with the different graduate disciplines, it would be fair to assume that the underlying samples
would be independent. To check for the population standard deviation, F test for variances
may be run as per which the p value comes out to be greater than 0.05 and hence it may be
assumed that the variances of the two populations would be equal. Thus, this assumption is
complied with in this case.

The requisite hypothesis is as highlighted below.
Null Hypothesis (H0): μCommerce = μHealthScience i.e. the average salary of the commerce graduates
and health science graduates do not differ significantly.
Alternative Hypothesis (H1): μCommerce < μHealthScience i.e. the average salary of the commerce
graduates is lower than the corresponding salary of health science graduates.
It is apparent from the above alternative hypothesis that the given test would be a left tailed t
test. The requisite computations are indicated below.
Let
Commerce (1)
n1=142,
s1=9247.05
x1=43298.54
Health Science (2)
n2 =107 ,
s2=10035.69
x2=41318.34
The value of pooled standard deviation is computed below:
SP= √ ( n1−1 ) s1
2+ ( n2−1 ) s2
2
n1 +n2−2
SP= √ ( 142−1 ) (9247.05)2+ ( 107−1 ) (10035.69)2
142+107−2
¿ √ 92033991.28
SP=9593.43

The value of test statistics is computed below:
t= x1 −x2
SP √ 1
n1
+ 1
n2
¿
( 43298.54−41318.34
9593.43∗
√ 1
142 + 1
107 )
t=1.6123
The value of degree of freedom is computed below:
Degree of freedom = n1 +n2−2=142+107−2=247
It is apparent that the value of the t statistic has come out to be positive while the rejection
region would lie on the negative side. Hence, it is evident that the null hypothesis would not
be rejected. Therefore, it is evident that the average salary levels of commerce and health
science graduates do not show any significant statistical difference.
Employment comparison (Law & Engineering)
For comparing the proportion of the employed graduates in law and engineering discipline,
the appropriate test would be Z test for comparison of two proportions which is based on the
approximation of binomial distribution as normal distribution. The various assumptions for
this test are as follows.
 Random sample should be used for sampling of the given samples.
 Samples must be independent.
 There should be inclusion of atleast 10 failures and same number of successes for
each sample.
 Also, np and n(1-p) must both be atleast 10 where n is the sample size and p is the
probability of success.
As has been discussed above the given samples are independent and also the sampling
method deployed is random only. Also, it is apparent that there are more than 10 successes
and failures for each of the samples which would be apparent from the test input values

highlighted below. Also, considering that n > 100 for each of the samples and p is not an
extreme value, np and np(1-p) for both the samples do exceed 10.
The requisite hypothesis is as highlighted below.
H0: plaw = pengineering
H1: plaw ≠ pengineering
The test statistic computation is shown below.
n1=121
p1= 81
121 =0.669
n2 =148
p2= 75
148 =0.507
Z value is computed below:
z=
{ ( p1−p2 )
√ p1 ( 1−p1 )
n1
+ p2 ( 1− p2 )
n2
}
¿ ( 0.669−0.507 )
√ 0.669 ( 1−0.669 )
121 + 0.507 ( 1−0.507 )
148
¿ 0.163
0.593
¿ 2.73
Considering a significance level of 5% and two tail test, the relevant critical value would be
+/- 1.96. Since the computed value of the z statistic does not lie in the critical interval, hence
the null hypothesis would be rejected and alternative hypothesis would be accepted. Hence,

the employment proportion of law and engineering tend to show statistically significant
difference.
Conclusion
Based on the above analysis, it may be concluded that the employment status of the graduates
belonging to the four disciplines do not exhibit any significant difference. However, the same
cannot be said about the salary levels which do tend to exhibit significant difference for
graduates of atleast one discipline. Further, the given sample data does not support the claim
that the salary levels of commerce graduates is lower than the health science graduates. Also,
there seems to be statistically significant difference in the proportion of graduates employed
from law and engineering disciplines. These conclusions would be useful to the career centre
in understanding the employment trends for the various disciplines and provide guidance to
the students accordingly.