| Course | MST-7000 Managerial Statistics |
|---|---|
| Module | Module 5 |
| Paper type | Module 5 written assignment: comparisons involving population means |
| Length | About 567 words, 5 pages |
| Format | APA 7 student paper |
| School | Thomas Edison State University |
| Program | MBA in Data Analytics |
| Updated | October 2026 |
Free sample paper for MST 7000 Module 5
Faster by Seven Tenths of a Day: Comparing Two Carriers' Delivery Times
Student Name
MBA in Data Analytics, Thomas Edison State University
MST-7000: Managerial Statistics
Mentor Name
Month Day, Year
Faster by Seven Tenths of a Day: Comparing Two Carriers' Delivery Times
The Question
After the probability memo showed that carrier A, the regional carrier, damaged more items, the operations director asked whether carrier B, the national carrier, is also faster. Carrier B charges $18 more per delivery. I sampled delivery times for 45 carrier A orders and 40 carrier B orders from the same eight weeks and the same delivery zones.
| Sample | n | Mean days | Standard deviation |
|---|---|---|---|
| Carrier A | 45 | 6.4 | 1.6 |
| Carrier B | 40 | 5.7 | 1.2 |
Choosing the Test
The samples are independent, since each order used one carrier. Anderson et al. (2011) present the t test for the difference between two population means when the population standard deviations are unknown. Excel's ToolPak offers versions that assume equal variances and that do not. The sample standard deviations differ, 1.6 against 1.2, so I used the unequal variance version, known as Welch's test (Welch, 1947). Ruxton (2006) argued that this test should be used routinely, because it performs well whether or not the variances are equal, while the equal variance test can mislead when they are not.
The Test
Because the p-value falls under 0.05, we reject the null hypothesis: the evidence indicates that the carriers' mean delivery times differ, with carrier B faster.
| Element | Value |
|---|---|
| Null hypothesis | Mean A minus mean B equals 0 |
| Alternative | Mean A minus mean B does not equal 0 |
| Significance level | 0.05 |
| Standard error of the difference | 0.305 days |
| t statistic, 0.7 / 0.305 | 2.30 |
| Degrees of freedom, Welch formula | 80.8 |
| p-value, two-tailed | 0.024 |
Checking Conditions
The t test needs each sample to come from a population that is roughly normal, or else to contain enough observations that the central limit theorem applies. With 45 and 40 orders, and histograms showing only mild right skew, the condition is reasonably met. I also checked that no single order dominated either mean; the longest delivery was 11 days for carrier A and 9 for carrier B, and removing both changed the difference by less than a tenth of a day.
The Size of the Difference
A 95 percent interval for the difference is 0.7 plus or minus 1.99 x 0.305, or 0.09 to 1.31 days. Carrier B is faster, but the true advantage could be as small as about two hours or as large as about a day and a third. Wasserstein and Lazar (2016) caution that statistical significance does not establish practical importance; the interval shows how uncertain the size of the advantage is.
Is It Worth It?
Shifting 300 deliveries a month to carrier B would cost $5,400 a month. If the advantage is near 0.7 days, about 14 percent more of those orders would arrive within the seven-day promise, based on the normal model from Module 3, and the probability memo suggests damage would fall as well. But if the true difference is near the low end, the speed gain alone would not justify the cost.
Limitations
The samples are observational: the carriers may serve slightly different addresses within the zones, and weather in the eight weeks affected both. A planned comparison, in which orders are randomly assigned to carriers for a month, would give a cleaner answer.
Recommendation
Shift the most damage-prone items, sofas and glass tables, to carrier B now, where both speed and damage favor it, and run a one-month randomized comparison before moving other volume.
References
Anderson, D. R., Sweeney, D. J., & Williams, T. A. (2011). Essentials of modern business statistics with Microsoft Excel (5th ed.). South-Western Cengage Learning.
Ruxton, G. D. (2006). The unequal variance t-test is an underused alternative to Student's t-test and the Mann-Whitney U test. Behavioral Ecology, 17(4), 688-690. https://doi.org/10.1093/beheco/ark016
Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129-133. https://doi.org/10.1080/00031305.2016.1154108
Welch, B. L. (1947). The generalization of Student's problem when several different population variances are involved. Biometrika, 34(1/2), 28-35. https://doi.org/10.2307/2332510
What the MST 7000 Module 5 instructions ask for
The fifth written assignment covers comparisons involving population means. Expect problems on the difference between two independent means, matched or paired samples, and possibly proportions. You will need to choose the right test, state hypotheses, run the test with Excel's ToolPak or functions, build an interval for the difference and interpret the result. Explain why you chose the equal or unequal variance version, and report the size of the difference along with the p-value, since managers decide on how large a difference is. Check the test's conditions, including the shape of each sample and the effect of extreme values, and report what you found. Keep the two samples in separate labeled columns.
How this MST 7000 Module 5 example is built
The memo opens with the question and a table of sample statistics. A section explains why the samples are independent and why the unequal variance test was chosen, with research support. A table sets out the test, and the text states the conclusion. A section builds an interval for the difference and explains what its width means. A cost comparison asks whether the difference is worth paying for, and limitations and a recommendation close the memo. A short section checks the test's conditions, the shape of each sample and the effect of removing the longest deliveries, before the size of the difference is discussed. The recommendation separates what can be done now from what needs a cleaner test first.
Reading the MST 7000 Module 5 grading rubric
Grading here weighs the correct choice of test, correct calculations, clear hypotheses and thoughtful interpretation. Papers that justify the unequal variance test, and that report an interval for the difference, show command of the module. Weighing practical importance against cost demonstrates managerial judgment. Noting that observational samples limit causal claims, and suggesting a better design, shows awareness of how data were gathered. Checking conditions, such as sample shape and the influence of extreme values, shows that the result was tested rather than assumed. Laying out the test's elements in a table lets a reader follow the reasoning and compare it with the ToolPak output.
MST 7000 Module 5 help: mistakes that cost marks
Students lose points by using a paired test for independent samples, by choosing the equal variance test without checking, by reporting only the p-value or by claiming one carrier causes faster delivery from observational data. Others misread the ToolPak's one-tail and two-tail outputs. Decide whether samples are independent or paired. Report the difference and its interval. Use the two-tailed p-value for a two-sided question. Ask whether the difference matters in money or time. Check whether one or two unusual values drive the difference. Run the test in the ToolPak, then confirm the t statistic by hand from the standard error so you understand where it comes from. A free first sample on your own MST-7000 comparison of means assignment is available on request.
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MST 7000 Module 5 questions, answered
Where can I find a free MST 7000 Module 5 sample paper?
Two carriers' delivery times compared with Welch's t test, for a furniture retailer, are worked in the assignment above.
When should the unequal variance t test be used?
When population variances may differ; research suggests it works well whether or not they are equal.
What is the difference between independent and paired samples?
Independent samples come from separate groups; paired samples measure the same units twice or match them.
Why report an interval for the difference?
It shows how large the difference could plausibly be, not only whether it is significant.
Which ToolPak tool runs Welch's test?
t-Test: Two-Sample Assuming Unequal Variances.