Mathematics • Year 7 • Unit 4 • Lesson 20
Data and Chance Synthesis — Mixed Challenge
Bring together PPDAC, relative frequency, expected frequency (E = P × n), and the difference between expected and actual. Spot a common conclusion mistake and plan your own data investigation.
1. Mixed problems
Show working. 2 marks each
1.1 A fair coin is flipped 500 times. How many heads are expected? Why might the actual count be 487 instead?
1.2 A spinner has P(red) = 0.25 and P(green) = 0.30. Find the expected number of red and green outcomes in 400 spins.
1.3 A survey of 240 students records "favourite season": Summer 96, Autumn 48, Winter 36, Spring 60. Find the relative frequency for each season and confirm they sum to 1.
1.4 Using the survey in 1.3, predict the favourite-season counts for a Year 7 cohort of 360 students.
1.5 A school knows that 12% of all students have a peanut allergy. In a Year 7 cohort of 250, how many students with peanut allergy are expected? Why is this only an estimate?
1.6 A PE teacher records mile-run times for 50 students. 22 ran the mile in under 8 minutes. (i) Find the experimental P(time < 8 min). (ii) If the same school has 600 students, predict how many would run a mile in under 8 minutes. (iii) Name the PPDAC stage of step (ii).
2. Find the mistake
A Year 7 student writes the following about a class survey of 20 students, where 4 students said they walked to school: Exactly one line contains a serious error of reasoning. Spot it, explain why it's wrong, then write the correct version. 3 marks
Student's working:
Line 1: Total students surveyed = 20. Walkers = 4.
Line 2: Experimental P(walks) = 4 ÷ 20 = 0.2.
Line 3: The school has 500 students. E(walkers) = 0.2 × 500 = 100.
Line 4: CONCLUSION: Exactly 100 students at this school walk to school each day, with no uncertainty, because we did the maths.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Rewrite Line 4 as a correct, careful conclusion.
Stuck? Revisit lesson § "Spot the Trap" — small samples + the word "exactly" almost always means the conclusion is overclaiming.3. Open-ended challenge — design your own PPDAC investigation
This question has many correct answers. Show your work clearly. 4 marks
3.1 Design a one-page PPDAC investigation for your class. You must give:
- Problem — a clear yes/no or how-many question about Year 7s at your school,
- Plan — sample size and the survey question or measurement (one sentence),
- Data — invent a small results table (4–6 rows) consistent with your plan,
- Analysis — compute one relative frequency AND one expected count for a larger group using E = P × n,
- Conclusion — answer your Problem question in one sentence, with the cautious wording the lesson recommends.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 500 fair coin flips
E = 0.5 × 500 = 250 heads. The actual count (487 heads from 1000 flips, or 247 from 500) often differs from the exact expected value because of natural variation — expected frequency is a prediction about a long-run average, not a guarantee for any single trial set.
1.2 — Spinner, 400 spins
E(red) = 0.25 × 400 = 100. E(green) = 0.30 × 400 = 120.
1.3 — Favourite season relative frequencies
P(Summer) = 96/240 = 0.40; P(Autumn) = 48/240 = 0.20; P(Winter) = 36/240 = 0.15; P(Spring) = 60/240 = 0.25. Sum = 0.40 + 0.20 + 0.15 + 0.25 = 1.00. ✓
1.4 — Predict for 360 students
E(Summer) = 0.40 × 360 = 144; E(Autumn) = 0.20 × 360 = 72; E(Winter) = 0.15 × 360 = 54; E(Spring) = 0.25 × 360 = 90. Sum = 360. ✓
1.5 — Peanut allergy
E = 0.12 × 250 = 30 students. It is only an estimate because expected frequency predicts an average — natural variation could push the actual count above or below 30 in any given cohort.
1.6 — Mile run
(i) P(under 8 min) = 22 ÷ 50 = 11/25 = 0.44.
(ii) E = 0.44 × 600 = 264 students.
(iii) Step (ii) is part of the Conclusion stage of PPDAC — using the analysis to answer a question about the wider population.
2 — Find the mistake
(a) The mistake is on Line 4.
(b) Expected frequency (100) is a prediction, not a guaranteed count. A small sample of only 20 students gives a noisy estimate, and natural variation means the actual school-wide count could be well above or below 100.
(c) Correct: "Based on a small sample of 20 students, we estimate that about 100 students at this school walk — but with only 20 surveyed, the true number could plausibly be anywhere between roughly 60 and 140. A larger survey would tighten this estimate."
3 — Design your own PPDAC investigation (sample answer)
Problem: What proportion of Year 7s at our school eat breakfast every weekday?
Plan: Survey 40 Year 7 students chosen at random. Ask: "Did you eat breakfast every weekday this past school week (yes/no)?"
Data (invented):
Response Frequency
Yes 28
No 12
Total 40
Analysis: P(eats breakfast every weekday) = 28 ÷ 40 = 0.70. Expected count for the full Year 7 cohort of 240 = 0.70 × 240 = 168 students.
Conclusion: Based on this sample, about 168 of the 240 Year 7 students (≈ 70%) eat breakfast every weekday — though with a sample of only 40, the true count is an estimate, not an exact figure, and a larger survey would make us more confident.
Marking: 1 mark each for clear Problem + Plan, sensible Data, correct Analysis (P and E both shown), and a cautious Conclusion that does not over-claim.