Mathematics • Year 8 • Unit 4 • Lesson 1
Collecting Data — Mixed Challenge
Pull together everything from Lesson 1: identifying sampling methods, calculating stratified samples, spotting bias, and designing fair surveys. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 1. Show your working. 3 marks each
1.1 Name the sampling method for each scenario:
(a) A magazine prints a survey inside the issue and asks readers to mail it back.
(b) A school selects 5 students from each year group at random for a focus group.
(c) A statistician uses a random number generator to pick 100 names from the electoral roll.
1.2 A national park has 240 visitors on a busy weekend. The rangers want a stratified sample of 30 visitors broken down by group: 120 families, 60 hikers, 60 cyclists. How many of each? Show the sampling fraction.
1.3 Explain in one or two sentences why "the bigger the sample, the more accurate the result" is NOT always true. Use the word "bias".
1.4 Rewrite each biased question as a fair, neutral one:
(a) "Don't you think the new uniform is ugly?"
(b) "How much do you love the canteen food?"
1.5 A Year 8 cohort has 150 students. A stratified sample of 45 students is taken. The sample is broken down by class: 8A (12), 8B (15), 8C (18). Check whether the sample is in proportion to the cohort if 8A has 40 students, 8B has 50 students and 8C has 60 students. Show your check.
1.6 Match each scenario to the bias type (self-selection, sampling, question-wording, or response):
(a) Students underreport hours spent gaming when asked by their teacher.
(b) An online survey link is only shared in a parenting Facebook group.
(c) The question reads: "Would you support a fair pay rise for hard-working teachers?"
2. Find the mistake
Another student attempted this stratified-sample problem. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: A school of 400 students is 60% junior and 40% senior. Take a stratified sample of 40.
Line 1: Junior count = 0.60 × 400 = 240. Senior count = 0.40 × 400 = 160.
Line 2: Sampling fraction = 400 ÷ 40 = 10.
Line 3: Junior sample = 240 ÷ 10 = 24. Senior sample = 160 ÷ 10 = 16.
Line 4: Check: 24 + 16 = 40 ✓
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full. (The numerical answer happens to come out the same — explain why.)
Stuck? The sampling fraction is sample size ÷ population, not the other way round. The student inverted it.3. Open-ended challenge — design your own survey
This question has many valid answers. 4 marks
3.1 Your job: design a survey to find out how Year 8 students at your school spend lunchtime. The Year 8 cohort has 200 students split evenly across 8 classes of 25 each.
Write up your design with the following:
(i) State the population.
(ii) Choose ONE sampling method (random, systematic, stratified, or convenience) and explain why it suits this study. If you use stratified, calculate the number from each class for a sample of 40.
(iii) Write THREE fair survey questions. At least one must be a multiple-choice question with non-overlapping options.
(iv) Identify ONE possible source of bias and explain how your design tries to reduce it.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Sampling methods
(a) Self-selection / convenience (volunteers mail it back — not random). (b) Stratified random sampling (equal numbers from each stratum — year group). (c) Random sampling (random number generator gives every member equal chance).
1.2 — National park stratified sample
Sampling fraction = 30 ÷ 240 = 1/8. Families: 120 × 1/8 = 15. Hikers: 60 × 1/8 = 7.5 — but we must round to whole people, so 8. Cyclists: 60 × 1/8 = 7.5 → 7. Check: 15 + 8 + 7 = 30 ✓ (Rounding is common in real-world stratified samples — adjust the largest groups to keep the total exact.)
1.3 — Size vs quality
A bigger sample is not automatically more accurate, because a large sample can still suffer from bias if it was chosen badly. For example, a survey of 5,000 gym members about exercise habits will overstate fitness levels, no matter how big it gets, because the sample is not representative of the whole population. Sample quality (random and representative) matters more than raw size.
1.4 — Rewrite biased questions (sample)
(a) "What is your opinion of the new uniform? Like it / Neutral / Dislike it / Unsure" — neutral wording, balanced options. (b) "How would you rate the canteen food? Very poor / Poor / OK / Good / Very good" — neutral, balanced 5-point scale with no leading wording.
1.5 — Check the Year 8 cohort stratification
Sampling fraction = 45 ÷ 150 = 0.30 (= 3/10). Expected: 8A → 40 × 0.3 = 12 ✓; 8B → 50 × 0.3 = 15 ✓; 8C → 60 × 0.3 = 18 ✓. The sample is correctly stratified — each class is sampled at 30%, matching the overall ratio.
1.6 — Match the bias type
(a) Response bias — students don't answer honestly to a teacher. (b) Sampling bias — the sample only reaches parents already in that group, not the wider population. (c) Question-wording bias — "fair" and "hard-working" lead respondents toward Yes.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The sampling fraction should be sample size ÷ population, not population ÷ sample. So it should be 40 ÷ 400 = 1/10, not 10. The student inverted the ratio.
(c) Corrected working:
Junior count = 0.60 × 400 = 240. Senior count = 0.40 × 400 = 160.
Sampling fraction = 40 ÷ 400 = 1/10.
Junior sample = 240 × 1/10 = 24. Senior sample = 160 × 1/10 = 16.
Check: 24 + 16 = 40 ✓
The final numbers happen to be the same as in the student's working because dividing by 10 gives the same result as multiplying by 1/10. But the student's reasoning was wrong — if the sampling fraction had been, say, 1/4, the inverted form (4) would give wildly different answers.
3 — Open-ended survey design (sample solution)
(i) Population: All Year 8 students at the school (200 students).
(ii) Method: Stratified random sampling by class. The cohort is split into 8 classes of 25, so stratifying by class guarantees every class is represented. Sampling fraction = 40 ÷ 200 = 1/5, so we randomly pick 5 students from each of the 8 classes (5 × 8 = 40 ✓).
(iii) Sample fair questions:
Q1: "What is your MAIN activity at lunchtime? Library / Sport on oval / Canteen / Sitting outside / Music room / Other ____" — non-overlapping single-choice options.
Q2: "How many minutes of your lunch break do you usually spend eating? Less than 5 / 5 to less than 15 / 15 to less than 30 / 30 or more" — neutral wording, non-overlapping intervals.
Q3: "Do you feel lunchtime is long enough? Yes / No / Unsure" — balanced options including a neutral choice.
(iv) Source of bias: Response bias — students might give the answer they think is "expected" by the teacher. Reducing it: collect responses anonymously (no names), and have a peer (not a teacher) hand out the survey, so respondents feel free to answer honestly.
Marking: 1 mark for correctly stating population. 1 mark for a justified sampling method with correct numbers if stratified. 1 mark for three fair questions with at least one non-overlapping MC. 1 mark for identifying a bias source AND a plausible mitigation.