Mathematics • Year 10 • Unit 4 • Lesson 20
Statistics & Probability in the Real World
Apply the full Unit 4 toolkit to real datasets: school test scores, a weather record, technology ownership, a sport injury study, and a class survey. Decide which technique fits each problem before computing.
1. Word problems
Pick the right Unit 4 tool: five-number summary, scatter plot, addition rule, multiplication rule, conditional probability. Show all working.
1.1 — Year 10 test scores. A Year 10 Mathematics class scored, out of 100: 58, 62, 65, 68, 70, 72, 74, 76, 78, 80, 82, 85, 88, 90, 95.
(a) Find the median, Q1, Q3 and IQR.
(b) Sketch a box plot (a quick number-line sketch is fine).
(c) The teacher wants to know "the typical mark". Which measure of centre would you recommend — mean, median or mode? Justify in one sentence. 3 marks
1.2 — BOM rainfall record. Daily Sydney rainfall over 7 days (mm): 0, 0, 2, 5, 12, 18, 60.
(a) Find the mean and median.
(b) Identify the value that is likely an outlier and explain in one sentence why the median is the better summary of "a typical day". 3 marks
1.3 — Technology in a Year 10 cohort. A survey of 100 students: 70 own a smartphone, 50 own a laptop, 35 own both.
(a) Build a two-way table (phone Y/N × laptop Y/N).
(b) Find P(phone or laptop) and P(neither).
(c) Find P(laptop | phone), and compare with P(laptop). Are owning a phone and owning a laptop independent? 3 marks
1.4 — School injuries. A study of 200 Year 10 students records two variables: plays rugby (Y/N) and visited the school nurse for an injury this term (Y/N). 80 play rugby; 25 of those visited the nurse. 30 non-rugby players visited the nurse.
(a) Build the two-way table with all four cells and totals.
(b) Find P(injury | plays rugby) and P(injury | doesn't play rugby).
(c) In one sentence, comment on whether rugby is associated with more nurse visits. 3 marks
1.5 — Coin-and-die game. A game involves flipping a fair coin and rolling a fair die. You "win" if you get a head AND a number greater than 4.
(a) State why the coin and die are independent in one sentence.
(b) Find P(win).
(c) You play 60 rounds. How many wins would you expect? 3 marks
2. Explain your thinking
Communication question. Use full sentences. 4 marks
2.1 A news article reports: "The average Sydney income is $95,000, so over half of Sydney workers earn more than $95,000." Write a four-sentence reply that (i) identifies the statistical error the article is making (confusing mean with median for a skewed distribution), (ii) explains in plain English why the mean income can be much higher than the median, (iii) names the Lesson 20 MCQ Q3 idea (which measure of centre is best when data has outliers), and (iv) finishes with a one-sentence rule of thumb for choosing mean vs median in news stories.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Year 10 test scores
(a) 15 values in order. Median = 8th value = 76. Lower half (first 7): 58,62,65,68,70,72,74 → Q1 = 4th = 68. Upper half (last 7): 78,80,82,85,88,90,95 → Q3 = 4th of upper = 85. IQR = 85 − 68 = 17.
(b) Box plot from 58 to 95, box from 68 to 85, median line at 76.
(c) No extreme outliers; the data is fairly symmetric. Either mean or median is reasonable — median is the safer choice and aligns with the MCQ Q3 rule.
1.2 — Rainfall
(a) Sum = 97. Mean = 97/7 ≈ 13.9 mm. Median (middle of 7) = 5 mm.
(b) Outlier = 60 (much larger than the other values). The mean (13.9) is pulled up by the single big-rain day; the median (5) better reflects what happens on a "typical" day.
1.3 — Smartphone and laptop
Phone only = 35, both = 35, laptop only = 15, neither = 15.
| Lap Y | Lap N | Total
Ph Y | 35 | 35 | 70
Ph N | 15 | 15 | 30
Tot | 50 | 50 | 100 ✓
(b) P(phone or laptop) = 85/100 = 0.85. P(neither) = 0.15.
(c) P(laptop | phone) = 35/70 = 0.5 = P(laptop). So phone and laptop ownership are independent in this sample.
1.4 — Rugby injuries
80 rugby, 120 non-rugby. 25 rugby + injury; 30 non-rugby + injury. So 80−25 = 55 rugby no-injury; 120−30 = 90 non-rugby no-injury.
| Inj Y | Inj N | Total
Rugby Y | 25 | 55 | 80
Rugby N | 30 | 90 | 120
Tot | 55 |145 | 200 ✓
(b) P(injury | rugby) = 25/80 = 0.3125. P(injury | not rugby) = 30/120 = 0.25.
(c) Rugby players are more likely to visit the nurse for an injury (0.31 vs 0.25), suggesting rugby is associated with a higher injury rate.
1.5 — Coin-and-die game
(a) The coin and die are mechanically independent — one cannot affect the other.
(b) P(head) = 1/2. P(> 4) = 2/6 = 1/3. P(win) = 1/2 × 1/3 = 1/6.
(c) Expected wins = 60 × 1/6 = 10.
2.1 — Mean vs median in the news (sample response)
The article confuses mean with median. Income distributions are heavily right-skewed: a small number of very high earners (executives, professionals) drag the mean up, while the median splits the population exactly in half. So a "mean income of $95,000" does not mean half of workers earn more than that — typically the median income is much lower, and well over half of workers earn below the mean. Lesson 20 MCQ Q3 captures this: when the data has outliers, the median is the better measure of centre. Rule of thumb: when a news story reports an "average", ask whether it's the mean or the median — for skewed data like income, house prices or rainfall, only the median answers "what is typical".
Marking: 1 mark per part (i)-(iv).