Mathematics • Year 7 • Unit 4 • Lesson 20
Data and Chance Synthesis — Real World
Apply the PPDAC cycle, relative frequency, and expected frequency to real investigations: a screen-time survey, a school sport tracker, a weather log, an absentee study, and a school canteen audit.
1. Word problems
Show formula lines (P = freq ÷ total; E = P × n).
1.1 — Screen-time survey. A sample of 30 Year 7 students records their daily screen-time. 18 of them report MORE than 3 hours. (a) Find the experimental probability that a Year 7 student exceeds 3 hours of screen-time. (b) If the school has 240 Year 7 students, estimate how many exceed 3 hours daily. 3 marks
1.2 — School sport tracker. A sport co-ordinator records the results of 60 inter-school matches: Wins 24, Draws 12, Losses 24. (a) Find P(win), P(draw), P(loss). (b) Predict how many wins, draws and losses are expected in a 100-match season at this rate. (c) Check that your three predictions add to 100. 4 marks
1.3 — Weather log. Over 200 school days, it rained at recess on 30 days. (a) Find the experimental P(rain at recess) on a school day. (b) Estimate the number of rainy recess days in a 40-day term. (c) Why is this estimate more reliable than one from only 20 days of data? 4 marks
1.4 — Absentee study. A school records absences over 180 school days for 600 students. Total absences = 5400. (a) Find the experimental probability that a randomly selected student-day is an absence. (b) On a typical school day, how many students would you expect to be absent? 3 marks
1.5 — Canteen audit. Out of 500 lunch orders one week: hot meals 200, sandwiches 150, salads 75, drinks-only 75. (a) Find the relative frequency for each category. (b) Predict the order counts for next month, which will have 2000 orders. (c) Use the PPDAC cycle to name which stage each of (a) and (b) belongs to. 4 marks
2. Explain your thinking
Communication matters. Use full sentences. 4 marks
2.1 A Year 7 student says: "P(win) = 0.4 and the team plays 10 games next month, so they WILL win exactly 4 games." Using the lesson, explain (i) what is wrong with the word "exactly", (ii) what E = 0.4 × 10 = 4 really means, and (iii) name two realistic outcomes (with their win counts) that would NOT be surprising in any single 10-game block.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Screen-time survey
(a) P(> 3 h) = 18 ÷ 30 = 3/5 = 0.6.
(b) E = 0.6 × 240 = 144 students.
1.2 — Sport tracker
(a) P(W) = 24/60 = 0.4; P(D) = 12/60 = 0.2; P(L) = 24/60 = 0.4.
(b) E(W) = 0.4 × 100 = 40; E(D) = 0.2 × 100 = 20; E(L) = 0.4 × 100 = 40.
(c) Sum = 40 + 20 + 40 = 100. ✓
1.3 — Weather log
(a) P(rain at recess) = 30 ÷ 200 = 3/20 = 0.15.
(b) E = 0.15 × 40 = 6 days.
(c) The 200-day sample is large enough to smooth out short-term weather streaks; with only 20 days a wet week could push the value far from the true long-run probability — the law of large numbers.
1.4 — Absentee study
(a) Total student-days = 600 × 180 = 108 000. P(absence) = 5400 ÷ 108 000 = 0.05 = 5%.
(b) Expected absences on one day = 0.05 × 600 = 30 students.
1.5 — Canteen audit
(a) P(hot) = 200/500 = 0.40; P(sand) = 150/500 = 0.30; P(salad) = 75/500 = 0.15; P(drinks-only) = 75/500 = 0.15. Sum = 1.00. ✓
(b) For 2000 orders: hot = 0.40 × 2000 = 800; sandwiches = 0.30 × 2000 = 600; salads = 0.15 × 2000 = 300; drinks-only = 0.15 × 2000 = 300.
(c) (a) is part of the Analysis stage (computing statistics from data). (b) is part of the Conclusion stage (using analysis to answer a prediction question about the future).
2.1 — Explain your thinking (sample response)
(i) The word "exactly" is too strong. Expected frequency gives a predicted average, not a guaranteed outcome — a 10-game block is a single small sample, and random variation almost always pushes the actual count above or below the predicted value.
(ii) E = 0.4 × 10 = 4 means that if the team were to play many separate 10-game blocks under the same conditions, the average number of wins per block would be about 4 wins.
(iii) Examples of unsurprising single-block outcomes: 3 wins (one below expected) or 5 wins (one above expected). Even 2 or 6 wins would not be shocking with only 10 games.
Marking: 1 mark for criticising "exactly", 1 for explaining E as a long-run average, 1 for two reasonable alternative counts, 1 for clear sentences.