a. Are there differences in the employment status among the four groups of graduates

b. Are there differences in income among the four groups of graduates

c. Is income of Commerce graduates lower than income of Health Science graduates

d. Is the proportion of employed graduates in Law discipline different from the proportion of employed graduates in Engineering discipline.

o What are the underlying assumptions Have you checked the assumptions If checking for normal distribution, consider to put up the histogram and discuss the shape.

If you found it is not normally distributed, keep doing the test with the underlying assumption, but need to discuss this as one of your limitations in the overall conclusion.

## Answer:

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 ba

sed 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 (H_{0}): µ_{health}= µ_{Commerce}= µ_{Law}= µ_{Engineering }i.e. the employment status across the various graduate discipline does not alter in a statistical significant manner.

Alternative Hypothesis (H_{1}): 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

The above value needs to be compared with the critical value of F.

Assuming a level of significance of 5% and df_{1}= (4-1) = 3 and df_{2}= (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 (H_{0}): µ_{health}= µ_{Commerce}= µ_{Law}= µ_{Engineering }i.e. the salary status across the various graduate discipline does not alter in a statistical significant manner.

Alternative Hypothesis (H_{1}): 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 The above value needs to be compared with the critical value of F.

Assuming a level of significance of 5% and df_{1}= (4-1) = 3 and df_{2}= (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.

Null Hypothesis (H_{0}): µ_{Commerce} = µ_{HealthScience} i.e. the average salary of the commerce graduates and health science graduates do not differ significantly.

Alternative Hypothesis (H_{1}): µ_{Commerce} < µ_{HealthScience} 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 The value of pooled standard deviation is computed The value of test statistics is Degree of freedom.

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.

H_{0}: p_{law} = p_{engineering}_{}

H_{1}: p_{law} ≠ p_{engineering}

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.

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