Annotated exemplar · AI HL · deliberately mid-band
Are arrivals at the school canteen random? A mid-band AI HL draft, annotated to show how to improve
Exemplar written by IB Math Revision for teaching — not a real student's IA, not moderated by the IB. Study it — don't reuse it: submitting or copying it is academic misconduct. The data and the student voice are illustrative, and the marks are our judgement. Schools check coursework with similarity software such as Turnitin.
A deliberately mid-band AI HL draft on Poisson arrivals, with an annotation on every problem: an invalid χ² test (expected frequencies below 5, a degree of freedom not lost for the estimated λ), P(X = x) used as a p-value, a Type I error assumed to be exactly 5%, and a time pattern the student never plots. The annotations quote the corrected results — which reverse both conclusions.
How this exemplar would be marked
| Criterion | Mark | Why |
|---|---|---|
| A · Presentation | 3/4 | Coherent and well organised, with a clear aim; but padded with a textbook definition, and the conclusion claims certainty the tests cannot give. |
| B · Mathematical communication | 3/4 | Mostly appropriate notation and a probability table, but the random variable is never defined, the population/sample variance is unclear, and the figure is unlabelled and treats a discrete variable as continuous. |
| C · Personal engagement | 2/3 | Some engagement: a real question from the student council and data the student collected, but no decisions of the student's own drive the analysis. |
| D · Reflection | 1/3 | Limited, generic reflection only at the end; results that should have been questioned (variance well above the mean) are accepted. |
| E · Use of mathematics | 3/6 | Relevant AI HL mathematics (Poisson model, χ² goodness of fit, a Poisson hypothesis test, Type I error), but the χ² test is invalid, the degrees of freedom are wrong, the p-value is calculated incorrectly, and both conclusions are reversed when corrected. |
| Total | 12/20 | About 12/20. The topic and data are good enough for a much higher mark — every lost mark is recoverable with the fixes in the annotations. |
Marks are our judgement of this teaching exemplar against the current criteria, explained criterion by criterion. They are not IB moderation results.
Excerpts with examiner annotations
Free sections are shown below with comments; the rest is in the full exemplar, available in the protected viewer with the IA package or a Pro plan.
Introduction
I am on the student council and one of the most common complaints we get is about the queue in the canteen at lunchtime. The canteen manager told us that the number of students arriving is random so there is nothing they can do about it. In class we learned about the Poisson distribution, which is used to model random events such as customers arriving at a shop, so I thought it would be interesting to see if it works for our canteen.
Aim: to find out if the number of students arriving at the canteen follows a Poisson distribution, and to use this to test whether the canteen is busier on Fridays (pizza day).
The Poisson distribution is a discrete probability distribution which gives the probability of a certain number of events happening in a fixed time interval, when the events happen at a constant average rate and independently of each other. It was introduced by the French mathematician Siméon Denis Poisson in 1837.
C Good: a genuine reason for the question (the student council, the manager's claim). How to improve: turn the manager's claim into the point of the exploration — what would it mean for the queue if arrivals were not random, and what could the canteen change?
A How to improve: the aim is clear but it is only a test (“does it follow…”), with no purpose after the answer. Add what you will do with the model, e.g. estimate how often more than a set number of students arrive in a minute.
A How to improve: the textbook definition and history add nothing, and they are paraphrased from an online source without a citation — unreferenced material like this is an academic-honesty risk, not just padding. Replace them with the two conditions that matter for your data — a constant rate and independence — and say whether you expect them to hold at a canteen.
Data collection
I stood by the canteen door and counted how many students joined the queue in each minute from 12:30 (when the bell goes) to 13:10, using the stopwatch on my phone. I did this on a Monday, Tuesday and Wednesday, which gave me 120 one-minute counts. The results are below.
Data note: these counts are illustrative, simulated by IB Math Revision for this annotated draft; a student must collect their own.
| Students per minute | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10+ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Frequency | 7 | 17 | 24 | 23 | 15 | 12 | 7 | 8 | 4 | 2 | 1 |
The mean number of students per minute is \(\bar{x} = 3.43\).
B How to improve: say what counts as “joining” (entering the door? joining the back of the queue?), and how you avoided losing count in a busy minute. Define the random variable: \(X\) = number of students joining the queue in one minute.
E How to improve: the counts are never plotted against time. Doing so shows the problem with the whole exploration: in the first ten minutes after the bell the mean is 5.00 per minute, and after that 2.91. The rate is not constant, which breaks a Poisson condition before any test is done.
D How to improve: pooling three days and 40 minutes into one table hides exactly the pattern that answers the manager's claim. Reflect on whether the data can be treated as one distribution before treating them as one.
The Poisson distribution
In the full exemplar (about 1.5 pages). The Poisson formula, the mean–variance check, and a table of probabilities and expected frequencies. Open in the protected viewer
Chi-squared goodness of fit test
In the full exemplar (about 1.5 pages). A χ² goodness-of-fit test for the Poisson model, with the degrees of freedom and the conclusion. Open in the protected viewer
Using the model
In the full exemplar (about 0.5 page). The Poisson model used to estimate how often the tills cannot keep up. Open in the protected viewer
Is it busier on Fridays?
In the full exemplar (about 1 page). A one-tailed Poisson hypothesis test for pizza day, and the probability of a Type I error. Open in the protected viewer
Conclusion
In conclusion, the number of students arriving at the canteen follows a Poisson distribution with mean 3.43 students per minute. The canteen is also busier on Fridays. This shows that the canteen manager is right that the arrivals are random, but they should add more staff on Fridays.
Limitations: I only collected data on four days and I may have made mistakes counting when it was busy. In the future I could collect more data and also look at how long each student waits in the queue.
A How to improve: both conclusions are wrong once the errors above are fixed, and the conclusion claims certainty (“follows”, “is busier”) that a hypothesis test cannot give.
D How to improve: the limitations are generic. The one that matters most — the arrival rate is much higher just after the bell — is never noticed. Reflect after each result, and say how each limitation affects your answer.
C How to improve: the recommendation (more staff on Fridays) does not follow from the mathematics. A stronger exploration would model the rush after the bell and use it to suggest something specific — e.g. staggering the release of year groups — and test that idea.
E How to improve: at AI HL, the extension is where the marks are: a model with a time-varying rate, or a queue simulation comparing staffing options, would lift Criterion E.
What would push it higher?
- Plot the counts against time before modelling; check the Poisson conditions (constant rate, independence) in context.
- Compare the variance with the mean quantitatively, and say what overdispersion would mean for the canteen.
- In a χ² goodness-of-fit test, merge classes so every expected frequency is at least 5, and subtract one degree of freedom for each parameter estimated from the data.
- Use P(X ≥ x) (or P(X ≤ x)) as the p-value, and calculate the actual Type I error probability for a discrete test.
- Compare like with like (the same time window), and extend the model — e.g. a time-varying rate — so the exploration answers the manager's claim.
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All 16 annotated exemplars, including three deliberately mid-band drafts.
Read the whole exploration
The full “Mid-band draft (Poisson, AI HL)” exemplar, with an examiner's note on every section, is in the IA package with the other 15 exemplars (Pro and Platinum plans include them too). €39 once, 12 months' access, 14-day money-back guarantee. It helps you write your own IA; it never writes it for you.
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