Mathematics Standard • Year 11 • Module 4 • Lesson 12
Module Review — Past-Paper Style
Practise HSC-style writing across the full module: three short-answer items (each drawn from a different lesson) and one structured long answer with explicit marking criteria.
1. Short-answer questions
1.1 The five-number summary for a sample is: min = 12, Q1 = 25, median = 30, Q3 = 40, max = 78. Use the 1.5 × IQR rule to determine whether 78 is an outlier. Show all working. 3 marks Band 3
1.2 The lifetime of a brand of LED bulb is approximately normal with mean = 8,000 hours and SD = 500 hours.
(a) Between what lifetimes do 95% of bulbs fall?
(b) Approximately what percentage of bulbs fail before 7,000 hours? 3 marks Band 3-4
1.3 A study reports r = 0.92 between number of Apple devices per household and average household income across 200 suburbs.
(a) Describe the correlation in plain English.
(b) Explain in 2 sentences whether this evidence shows that buying Apple devices raises household income. 4 marks Band 4
2. Extended response — full-module integration
2.1 A Year 11 PE department records, for 60 students, weekly exercise hours (x) and resting heart rate in beats per minute (y, dependent variable). The data range was 0 ≤ x ≤ 10 hours per week.
Reported statistics:
• Heart rate is approximately normal with mean = 70 bpm, SD = 8 bpm.
• Correlation between exercise hours and heart rate: r = −0.76.
• A line of best fit passes through (0, 75) and (8, 59).
(a) Describe the strength and direction of the reported correlation.
(b) Find the equation of the line of best fit, then use it to predict a student's resting heart rate at 6 hours/week of exercise.
(c) Using the normal distribution, estimate how many of the 60 students have a resting heart rate above 86 bpm.
(d) The PE teacher claims: "This data proves exercise lowers heart rate — students who don't exercise will have unhealthy hearts." Write a balanced 3-sentence response that (i) acknowledges the strong correlation, (ii) names ONE lurking variable that could be at play, and (iii) explains why a controlled experiment would be needed to claim causation. 8 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — describes r = −0.76 as a strong negative correlation.
Part (b) — 2 marks
• 1 mark — correct line equation y = −2x + 75.
• 1 mark — correct prediction at x = 6: y = 63 bpm (interpolation).
Part (c) — 2 marks
• 1 mark — recognises 86 = 70 + 2(8) = mean + 2 SD, and uses 2.5% upper tail.
• 1 mark — correct count: 60 × 0.025 = 1.5, i.e. about 1 or 2 students.
Part (d) — 3 marks
• 1 mark — acknowledges the strong negative correlation with statistic.
• 1 mark — names one plausible lurking variable (genetics, diet, baseline fitness, sleep, socioeconomic).
• 1 mark — proposes a controlled experiment (random assignment of non-exercisers to an exercise programme vs control) to establish causation.
Your response:
Stuck on (d)? Use the structure: acknowledge → identify lurking variable → propose controlled experiment.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Outlier check using 1.5 × IQR (3 marks)
Sample response. IQR = 40 − 25 = 15. Upper fence = 40 + 1.5(15) = 40 + 22.5 = 62.5. Since 78 > 62.5, 78 is an outlier.
Marking notes. 1 mark — IQR. 1 mark — upper fence calculation shown. 1 mark — conclusion comparing 78 to the fence and stating "outlier". A bare "yes, outlier" without working scores 1/3.
1.2 — LED bulb lifetimes (3 marks)
Sample response. (a) 8,000 ± 2(500) = 7,000 to 9,000 hours. (b) 7,000 = 8,000 − 2(500) = mean − 2 SD. Below 2 SD = (100 − 95) ÷ 2 = 2.5%.
Marking notes. (a) 1 mark — substitution and range with units. (b) 2 marks — 1 for identifying 7,000 as mean − 2 SD, 1 for 2.5% from the rule.
1.3 — Apple devices vs income (4 marks)
Sample response. (a) r = 0.92 → very strong positive linear correlation: suburbs with higher average household income tend to have more Apple devices per household. (b) This evidence does not show that buying Apple devices raises household income. The much more likely explanation is reverse causation: wealthier households can afford more premium devices. Additionally, lurking variables (education level, employment in tech-heavy professions) likely drive both income and device ownership.
Marking notes. (a) 1 mark — direction + strength. (b) 3 marks — 1 for rejecting causation, 1 for naming reverse causation explicitly, 1 for naming at least one lurking variable.
2.1 — Exercise vs heart rate (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Strength and direction.
r = −0.76 is a strong negative linear correlation: students who exercise more hours per week tend to have a lower resting heart rate. [1 mark.]
(b) Line of best fit + prediction at x = 6.
m = (59 − 75) ÷ (8 − 0) = −16 ÷ 8 = −2. b = 75 (from (0, 75)). y = −2x + 75. [1 mark.]
At x = 6: y = −2(6) + 75 = 63 bpm — interpolation (6 is within the 0–10 hour data range, so this is reliable). [1 mark.]
(c) Students above 86 bpm.
86 = 70 + 2(8) = mean + 2 SD. By the 68-95-99.7 rule, above 2 SD = (100 − 95) ÷ 2 = 2.5%. [1 mark.] 60 × 0.025 = 1.5 — about 1 or 2 students have a resting heart rate above 86 bpm. [1 mark.]
(d) Response to the PE teacher.
The data does show a strong negative correlation (r = −0.76) between weekly exercise and resting heart rate — students who exercise more tend to have lower resting heart rates. [1 mark — acknowledges with statistic.] However, this is observational data and a lurking variable such as baseline cardiovascular fitness (or genetics, or diet) could explain both who chooses to exercise and who has a low heart rate. [1 mark — lurking variable.] To claim that exercise causes a lower heart rate, the PE department would need a controlled experiment in which currently sedentary students are randomly assigned to an exercise programme or a control group, with heart rate measured before and after — only then can other explanations be ruled out. [1 mark — controlled experiment.]
Total: 8/8.
Band descriptors for marker.
Band 3: (a) correct; (b) correct equation OR correct prediction (not both); (c) attempts but misses 2.5%; (d) generic "correlation is not causation" without naming a lurking variable. ≈ 3-4 marks.
Band 4: All of (a), (b), (c) correct; (d) names lurking variable but doesn't propose an experiment. ≈ 5-6 marks.
Band 5: All numerical parts correct; (d) names lurking variable AND mentions "experiment" but without random assignment detail. ≈ 7 marks.
Band 6: All parts complete: r interpreted; line and prediction correct; 86 bpm correctly tied to mean + 2 SD with student count; (d) acknowledges with statistic, names a specific lurking variable, AND describes a controlled experiment with random assignment. 8/8.