Mathematics Standard • Year 11 • Module 4 • Lesson 12
Module Review — Problem Set
Apply the full Module 4 toolkit to five integrated, real-world scenarios — each draws on multiple lessons (data type, summary stats, normal distribution, scatter plots, correlation, line of best fit).
Problem 1 — Staffroom commute survey
A school surveys 50 staff. Daily commute times (minutes, one way) are recorded; the data is approximately normal with mean = 32 min and SD = 8 min. The principal also notes the five-number summary: min = 10, Q1 = 26, med = 32, Q3 = 38, max = 65.
Set up: What are we solving for?
(i) Compute the IQR. Use the 1.5 × IQR rule to check whether 65 is an outlier. 2 marks
(ii) Using the 68-95-99.7 rule, between what times do 95% of staff commute? 1 mark
(iii) The principal wants to know how many staff have a commute longer than 48 minutes. Estimate this percentage and convert it to a number out of 50. 2 marks
Stuck on (iii)? 48 = 32 + 2(8) = mean + 2 SD. Use the rule to find the upper tail.Problem 2 — Comparing house prices in two Sydney suburbs
A real-estate report (40 sales per suburb in 2025):
Suburb K: mean = $1.40 M, SD = $0.10 M; median = $1.42 M; IQR = $0.18 M.
Suburb L: mean = $1.40 M, SD = $0.40 M; median = $1.32 M; IQR = $0.45 M.
Set up: What are we solving for?
(i) Compare the two suburbs by centre (mean & median) and spread (SD & IQR). 2 marks
(ii) Which suburb shows evidence of skew? Justify using the relationship between mean and median. 2 marks
(iii) A buyer with a strict $1.20 M–$1.55 M budget asks which suburb is more "predictable" for their search. Recommend a suburb in one sentence. 2 marks
Stuck on (ii)? In a right-skewed distribution the mean is pulled above the median by high outliers.Problem 3 — Tutoring hours and HSC trial mark (multi-step)
A Sydney tutoring company tracks 12 students (weekly tutoring hours, HSC trial mark):
(0, 50), (1, 55), (2, 60), (2, 62), (3, 65), (3, 68), (4, 72), (4, 70), (5, 78), (5, 75), (6, 82), (6, 80)
Set up: What are we solving for?
(i) Describe the scatter (direction, strength, form). 2 marks
(ii) A line of best fit is drawn through (0, 50) and (6, 80). Find its equation. 2 marks
(iii) Predict the trial mark for 4 hours of tutoring (interpolation) and for 12 hours of tutoring (extrapolation). For the second, state two reasons the prediction is not trustworthy. 3 marks
Stuck? Revisit lesson § Line of Best Fit — calculate, predict, then classify.Problem 4 — Battery life of a power-tool brand
The Brunswick PowerPack 5.0 Ah batteries have life (in cycles before significant capacity loss) approximately normal with mean = 1,000 cycles and SD = 80 cycles.
Set up: What are we solving for?
(i) Between what cycle counts do 95% of batteries last? 1 mark
(ii) A warranty covers replacement of any battery failing before 840 cycles. Estimate the percentage of batteries that will be replaced under warranty. 2 marks
(iii) If the company makes 20,000 batteries per year, estimate the number replaced under warranty. Then suggest one design change that would reduce this number (referring to mean or SD). 3 marks
Stuck on (ii)? 840 = 1000 − 2(80) = mean − 2 SD; below 2 SD = 2.5%.Problem 5 — Critiquing a press release
A press release reads: "A new study of 1,000 NSW homes finds a strong positive correlation (r = 0.81) between number of indoor pot plants and self-reported well-being. Plants make us happy — buy a fern!"
Set up: What are we solving for?
(i) Describe the correlation in plain English. 1 mark
(ii) Identify two lurking variables that could explain the correlation. 2 marks
(iii) Propose one possibility of reverse causation, and write a one-sentence headline that would be a responsible report of this evidence. 3 marks
Stuck? Revisit lesson § Common Mistakes — "correlation = causation" is the most common error.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Staffroom commute
Set up. Combine IQR/outlier check with the 68-95-99.7 rule.
(i) IQR = 38 − 26 = 12. Upper fence = 38 + 1.5(12) = 38 + 18 = 56. 65 > 56, so 65 is an outlier.
(ii) 95% lie in 32 ± 2(8) = 16 to 48 minutes.
(iii) 48 = mean + 2 SD → above 2 SD ≈ 2.5%. 2.5% of 50 = 1.25 staff — about 1 staff member.
Problem 2 — Two suburbs
Set up. Use centre, spread and the mean-vs-median relationship to detect skew and to advise a buyer.
(i) Centre: identical means ($1.40 M); medians close (K $1.42 M, L $1.32 M). Spread: Suburb K is much tighter (SD $0.10 M vs $0.40 M; IQR $0.18 M vs $0.45 M).
(ii) Suburb L shows skew. Its mean ($1.40 M) is above its median ($1.32 M), which indicates right skew — a few expensive sales pull the mean up while the median sits with the typical home.
(iii) Suburb K is more predictable — its IQR ($0.18 M) sits comfortably inside the buyer's $0.35 M window ($1.20–$1.55 M), so most homes will be in range.
Problem 3 — Tutoring vs trial mark
Set up. Describe, fit, predict, and critique extrapolation.
(i) Strong positive linear correlation between weekly tutoring hours and trial mark; points cluster tightly to an upward line with no outliers.
(ii) m = (80 − 50) ÷ (6 − 0) = 30 ÷ 6 = 5. b = 50 (from (0, 50)). y = 5x + 50.
(iii) At x = 4: y = 5(4) + 50 = 70 marks (interpolation, reliable). At x = 12: y = 5(12) + 50 = 110 marks. Two reasons not trustworthy: (1) extrapolation — 12 hours is well outside the 0–6 hour data range; (2) marks cap at 100, so 110 is impossible.
Problem 4 — Battery life
Set up. Apply the 68-95-99.7 rule to a warranty problem.
(i) 1,000 ± 2(80) = 840 to 1,160 cycles.
(ii) 840 = mean − 2 SD → below 2 SD = (100 − 95) ÷ 2 = 2.5%.
(iii) 20,000 × 0.025 = 500 batteries/year replaced. Design change: reduce the SD (tighter manufacturing tolerances). With SD = 40, the warranty cut (840 cycles) would be at 1000 − 4 SD — extremely rare (< 0.005%) — almost no warranty claims. Alternatively, increase the mean to 1,100 cycles to shift the whole distribution above the cut.
Problem 5 — Pot plants and well-being
Set up. Critique a causal claim built from observational correlation.
(i) Strong positive linear correlation between number of indoor pot plants and self-reported well-being.
(ii) Lurking variables: (1) household income — wealthier households can afford plants and tend to report higher well-being for many reasons; (2) time at home and personality — people who enjoy nurturing and being at home may both keep plants and report higher well-being.
(iii) Reverse causation: happier people may be more likely to take on the work of buying and caring for pot plants — well-being causes plant ownership, not vice versa. Responsible headline: "Survey finds NSW homes with more indoor plants also report higher well-being — cause unclear."