| Course | BU 321 Business Statistics |
|---|---|
| Unit | Unit 3 |
| Paper type | Sampling and error write-up |
| Length | About 1,100 words, 6 pages |
| Format | APA 7 student paper |
| School | Herzing University |
| Program | BS in Business Management |
| Updated | September 2026 |
Free sample paper for BU 321 Unit 3
Who Would Pay $2 to Know When the Pizza Arrives? A Sampling and Error Write-Up for a Composite Delivery Chain's Customer Survey
Student Name
Business, Herzing University
BU 321: Business Statistics
Instructor Name
February 18, 2026
Who Would Pay $2 to Know When the Pizza Arrives? A Sampling and Error Write-Up for a Composite Delivery Chain's Customer Survey
The Question and Its Population
Brickoven Express, this course's made-up six-store pizza business, is considering a $2 option that would guarantee delivery within a 30-minute window or refund the fee. Before building it, the owners want to know what share of their delivery customers would choose it. The population this question implies is every household that has ordered delivery from Brickoven in the last 90 days, whether by app, website or phone. That population is not the same as the list of people the chain can easily reach, and the difference between the two is the first source of error this write-up addresses.
The Sampling Frame
A sampling frame is the list from which a sample is actually drawn. Brickoven's most practical frame is its list of customer email addresses collected from app and website orders in the last 90 days. In the order data described in Unit 1, 40.8 percent of orders were paid through the app and 47.1 percent by card, most of them on the website, while 12.1 percent were paid in cash, largely on phone orders. The email frame therefore covers customers responsible for roughly 88 percent of orders and misses most cash and phone customers. Groves et al. (2009) call the gap between a frame and its target population coverage error, and they note that it cannot be reduced by drawing a larger sample from the same frame.
The missing group matters for this question. Customers who phone in orders and pay cash may be older, less comfortable with apps or more price sensitive, and any of those traits could affect whether they would pay a $2 fee. The frame's results will describe app and website customers well and phone customers poorly.
The Sampling Method
The recommended method is stratified random sampling by store. The frame is divided into six strata, one per store, and an equal random sample is drawn from each. Stratifying guarantees that every store is represented, which matters because stores differ in delivery performance and possibly in customer attitudes toward a guarantee; a simple random sample might by chance draw few customers from a smaller store. Cochran (1977) showed that stratification reduces the variance of estimates when the strata differ from one another, which is likely here. The trade accepted is complexity: results must be weighted by each store's share of customers if the owners want a chainwide figure, since equal samples from unequal stores would otherwise overweight the small ones.
Convenience sampling, such as asking customers who happen to open the app on a given evening, was rejected. It would be fast and cheap, but it would oversample frequent users, who may value a guarantee differently from occasional customers, and the size of that bias could not be estimated.
How Large a Sample, and What Happens as It Grows
When the estimate is a proportion, like the fraction of customers who would pay the fee, the standard error of the sample proportion is the square root of p times (1 minus p) divided by n. Planning with p = 0.50, the value that gives the largest error, a sample of 100 completed responses gives a standard error of 0.050 and a 95 percent margin of error of about 9.8 percentage points. A sample of 400 gives a standard error of 0.025 and a margin of about 4.9 points. Quadrupling the sample halves the margin, because the standard error shrinks with the square root of n (Anderson et al., 2020). The recommendation is 400 completed responses, about 67 per store. If about one customer in four responds to an email survey, Brickoven should invite about 1,600 customers, roughly 267 per store.
The same pattern holds for the sampling distribution of a mean. Suppose the survey also asks for a typical order value. In the Unit 1 data, order value had a standard deviation of $10.79. If the population standard deviation is similar, the standard error of the sample mean would be $2.16 for 25 responses, $1.08 for 100 and $0.54 for 400. The mean of all possible sample means equals the population mean, so the sample mean is unbiased, and its spread shrinks steadily as n grows.
| Completed responses (n) | SE of a proportion at p = 0.50 | 95% margin (points) | SE of mean order value |
|---|---|---|---|
| 25 | 0.100 | 19.6 | $2.16 |
| 100 | 0.050 | 9.8 | $1.08 |
| 400 | 0.025 | 4.9 | $0.54 |
What the Central Limit Result Does and Does Not Promise
The central limit theorem states that the sampling distribution of the mean approaches a normal distribution as the sample size grows, whatever the shape of the population, provided the population has a finite variance (National Institute of Standards and Technology, 2012). Order value in the Unit 1 data had a skewness of only 0.28, so the sampling distribution of its mean would be close to normal even at 25 responses; a more skewed variable, such as delivery distance at 0.72, would need somewhat larger samples. This is what allows the margins in the table to be computed with normal critical values.
The theorem does not promise that the sample mean is close to the true value if the sample is biased. It describes the behavior of random samples drawn from the frame. If the frame leaves out phone customers, or if only enthusiastic customers respond, the sampling distribution will be centered on the wrong value, and a larger sample will only make the wrong answer look more precise.
Sampling Error Versus Frame and Response Error
Three kinds of error could affect the result. Sampling error is the chance difference between a random sample and the frame it came from; it is measured by the margins above and shrinks as n grows. Coverage error comes from the frame missing most phone and cash customers; it does not shrink with n. Nonresponse error arises if customers who answer differ from those who do not, for example if customers who have recently had a late delivery are more likely to reply; it also does not shrink with n (Groves et al., 2009).
Two steps would reduce the errors that sample size cannot fix. First, the chain could add a short phone version of the survey for a random sample of 60 phone-order customers, drawn from phone numbers on recent orders, to check whether their answers differ. Second, it could compare respondents with the frame on known facts, such as store and order frequency, and weight the results if respondents are unrepresentative. With those steps, a margin of about 5 points would be an honest description of the uncertainty rather than an understatement of it. Brickoven Express and its data are composites created for this course.
References
Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J. (2020). Statistics for business and economics (14th ed.). Cengage.
Cochran, W. G. (1977). Sampling techniques (3rd ed.). Wiley.
Groves, R. M., Fowler, F. J., Jr., Couper, M. P., Lepkowski, J. M., Singer, E., & Tourangeau, R. (2009). Survey methodology (2nd ed.). Wiley.
National Institute of Standards and Technology. (2012). NIST/SEMATECH e-handbook of statistical methods. https://www.itl.nist.gov/div898/handbook/
The Herzing BU 321 Unit 3 instructions
The Unit 3 write-up asks you to design a sample for a business question and explain the error involved. Most versions want the population defined, the sampling frame described with who is in and out of it, a sampling method chosen and justified, a discussion of the sampling distribution, including how its spread changes with sample size, and an explanation of what the central limit theorem allows. Many rubrics now ask you to distinguish sampling error from errors caused by a poor frame or low response. A worked calculation of standard error for one or two sample sizes is common. Keep the paper short and methodical, with the textbook and any survey methods source cited in APA 7.
Inside the BU 321 Unit 3 example
The sample states the business question and the population it implies, then describes the email frame and uses order data to estimate that it covers about 88 percent of orders. The method section argues for stratification by store and rejects convenience sampling with a reason. A sample size section works the standard error of a proportion and of a mean at three sizes and recommends 400 completed responses from about 1,600 invitations. The central limit section explains why the sampling distribution will be close to normal and warns that a biased frame shifts its center. The final section separates three kinds of error and proposes a phone check and weighting.
Where the points sit in the BU 321 Unit 3 rubric
Herzing's rubric for this write-up generally scores the definition of population and frame, justification of the method, the sampling distribution work and the treatment of error. Full points on population and frame require naming who the frame leaves out and why that matters for the question. The method criterion rewards a reason tied to the question and an honest statement of the trade accepted. Sampling distribution work is judged by correct standard errors and a clear explanation of how spread changes with n. The error criterion, increasingly important, rewards papers that separate sampling error from coverage and nonresponse error, as the sample's final section does. Clear tables and APA citations complete the grade.
BU 321 Unit 3 help before you submit
The most frequent Unit 3 weakness is treating the frame as if it were the population, which hides coverage error. Another is naming a sampling method with no argument for why it fits this particular question. Students often claim that a large sample fixes everything, when a larger sample from a biased frame only produces a more precise wrong answer. The central limit theorem is commonly overstated as a guarantee of accuracy rather than a statement about the shape of the sampling distribution. Some papers compute a standard error but never explain what it means. Finally, account for response rates; planning for completed responses rather than invitations is a practical detail instructors notice.
Get a BU 321 Unit 3 example written to your instructions
Send the Unit 3 instructions and the rubric from your BU 321 classroom, with the scenario or dataset the assignment names. We write a custom example that moves from question to population to frame, defends the method against that frame, shows the spread shrinking and separates the two errors. First custom sample free, returned in 24 to 48 hours.
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BU 321 Unit 3 questions, answered
How large should the sample in the example be?
Whatever the instructions set, and where they set nothing, choose a size and defend it from what the write-up needs to show. A number chosen because it is round invites the question of why. If the assignment asks you to justify a size, tie it to the precision you want rather than to convenience, and say what you traded away.
Is a convenience sample always wrong?
No, and pretending otherwise weakens a paper. Plenty of business data is collected that way, and the honest move is to name the method, say who it systematically misses and describe how that would bend the result. Sections generally mark a clearly labeled convenience sample with its limits stated above an unlabeled one dressed up as random.
What does the central limit result actually let me do?
It lets you treat the distribution of the sample mean as approximately normal under conditions your reading will state, which is what makes the later intervals and tests possible. What it does not do is make a small biased sample behave, or say anything about the shape of the underlying data. Name the conditions you relied on.
Where can I find a free BU 321 Unit 3 sample paper?
This page carries the finished BU 321 Unit 3 write-up free, with its sample size table and error discussion. To get one designed for your own business question, pass along the instructions and a free custom sampling plan will be written, generally within about two days.
Does a bigger sample fix bias in BU 321 Unit 3?
No. A larger sample shrinks sampling error, but coverage and nonresponse error stay the same. The sample shows the margin falling from about 10 points to 5 as responses grow from 100 to 400, then explains why missing phone customers would still bias the result.