Fun runs and t-shirts
A major charity organisation is organising a very large fundraising event in your city, the "City to Beach" fun run. They are expecting to have 65000 entrants in the inaugural event. To raise the profile of the event, every competitor who enters the fun run will receive a promotional t-shirt.
You have been put in charge of ordering an appropriate number of t-shirts in a suitable range of sizes. This is a considerable problem–if you order too few, the fun run entrants will be unhappy if they miss out on a t-shirt that is the correct size, but if you order too many, this could significantly impact the funds raised and we could end up trying to deal with hundreds of unwanted t-shirts.
A size chart from the t-shirt manufacturer is given here:
You will need to do your research to obtain any other information and data that you need to order the correct amount of t-shirts in the correct sizes.
Collecting data
You will need to decide how to collect data that can be used for your investigation.
The most obvious option is to ask each competitor to select their preferred shirt size when they enter the fun run. Unfortunately, this is not possible because it will not allow enough time for the t-shirts to be manufactured, printed and delivered.
Therefore you must consider obtaining data from other sources.
Question
2. What data do we require to be able to order the correct amount of t-shirts?
3. Consider the data required. Is it easily obtainable?
4. Do we need to make some assumptions to simplify the problem?
5. If we only had data on height or weight but not paired data, which would best to use to assess the number of t-shirts of different sizes required? Why?
6. What sources may we consider to be reliable?
One possible source of information is the Australian Bureau of Statistics which has data on the percentage of the Australian population estimated to be between certain weights. The data is for adults only and was voluntary and self-reported.
You could use the breakdown of participants' measured weights shown here.
| Measured weight (kg) | Males (%) | Females (%) |
|---|---|---|
| Less than 50 | 0.2 | 6.8 |
| 50 to < 60 | 3.8 | 27.2 |
| 60 to < 70 | 15.6 | 32.9 |
| 70 to < 80 | 28.8 | 18.6 |
| 80 to < 90 | 27.0 | 8.4 |
| 90 to < 100 | 14.7 | 3.5 |
| 100 to < 110 | 6.7 | 1.5 |
| 110 or more | 3.1 | 1.0 |
| Total | 100.0 | 100.0 |
Analysing and interpreting
This is the stage of your investigation where we "do the maths". It is important that you work carefully and systematically to ensure that your results and conclusions are accurate.
Questions
7. Form a list of assumptions made to utilise the data. Such as:
- Assuming there are an equal number of male and female runners, that is a population of 32500 male and 32500 female competitors.
- Assuming competitor's weights are distributed similarly to the overall population.
8. Use the data together with your assumptions to estimate the number of shirts required for each size and record the results in a table similar to that shown here. (Show the weight interval for each size in the first row)
| Weight | |||||||
|---|---|---|---|---|---|---|---|
| Size | XS | S | M | L | XL | XXL | XXXL |
| Male | |||||||
| Female | |||||||
| Total |
Now you have determined the quantities of each size of t-shirt that you need to order.
Often it is a good idea to construct a graph to represent our results. This is an excellent way to see for ourselves if the results appear to be reasonable and can also be used when we want to communicate our results and conclusions.
Questions
9. Create a histogram using the data in your table.
10. Using mathematical terminology, how could you describe the distribution of t-shirt sizes resulting from your calculations? Comment on the skew or symmetry, gaps, outliers or any other important information.
Now that we have produced our results, we need to consider if this is sufficient for your needs, if your assumptions are valid, or if you need to further refine our methods to get more accurate results.
Questions
11. Do the results of our calculations enable us to accurately order the correct amount of t-shirts?
12. Were the assumptions made reasonable? If not, what should have been changed?
13. Would another measurement be better to use than weight to select the best t-shirt size (e.g. height, Body Mass Index)? Justify your proposal.
14. How would you get more accurate information about the typical age and gender of fun-run participants? Explain your ideas.