Mathematics • Year 10 • Unit 4 • Lesson 2
Collecting Data — Skill Drill
Build fluency with the sampling vocabulary and the four methods from Lesson 2: random, stratified, systematic and convenience. Practise telling population from sample, and spotting bias — the lesson's warning that "a larger sample alone does not remove bias".
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. A principal of a school of 800 students wants to know how many minutes per night Year 10s spend on homework. They survey 40 students who are waiting in the front office at lunch on Monday. Name the sampling method, identify the population and sample, and explain whether the sample is likely to be biased.
Step 1 — Identify the population.
Population = the entire group of interest = all 800 students at the school.
Reason: Lesson 2 Key Term — "the entire group about which we want to draw conclusions".
Step 2 — Identify the sample.
Sample = the 40 students who happened to be in the front office at lunch.
Reason: a sample is a subset of the population selected to represent the whole.
Step 3 — Name the sampling method.
Surveying whoever happens to be available = convenience sampling.
Reason: from the Lesson 2 list of four methods, this matches "convenience" — easy to access, no random rule.
Step 4 — Decide on bias.
Students in the office at lunch may be in trouble, late, or skipping class.
Reason: they are not typical of all students. Lesson 2 — bias = a systematic error that misrepresents the population.
Answer: Population = 800 students, Sample = 40 office students, Method = convenience sampling, and the result is likely biased because office-loiterers are not representative of the school as a whole.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. A school of 1,200 students has 400 in each of Years 10, 11 and 12. A researcher randomly selects 30 students from Year 10, 30 from Year 11, and 30 from Year 12 to survey about screen time. Identify the population, sample, sampling method, and explain why this method usually reduces bias.
Step 1 — Population. The population is the __________________ students in the school.
Step 2 — Sample.
Sample size = 30 + 30 + 30 = ______ students.
Step 3 — Name the method. Splitting the population into year-group subgroups and randomly sampling each subgroup is __________________________ sampling.
Step 4 — Why it reduces bias. Each year group is __________________ in the sample in proportion to the population, so opinions of younger and older students are __________________ instead of being lost.
Step 5 — Sanity check. If the school had 600 in Year 10 and 300 each in Years 11 and 12, would equal samples of 30 per year still be fair? __________________ — because Year 10 would be __________-represented relative to the population.
3. You do — independent practice
For each scenario, name the sampling method used and whether the sample is likely to be biased. Then for the extension questions, explain what would be a better approach.
Foundation — name the method
3.1 A teacher puts every student's name in a hat and draws 20 names. Method = ? 1 mark
3.2 A researcher surveys every 10th customer leaving Woolworths until 50 surveys are filled in. Method = ? 1 mark
3.3 A reporter interviews their friends and family about a new bus route. Method = ? 1 mark
3.4 The ABS plans the Census so that every Australian household responds. Is this a sample or a census? 1 mark
Standard — name the method AND identify bias
3.5 A radio station asks listeners to phone in and vote on whether daylight saving should be abolished. After two hours they have 4,000 callers. Name the sampling method and explain why the result is biased, even though 4,000 is a large sample. 2 marks
3.6 A school of 600 students has 250 in junior years (Y7-9) and 350 in senior years (Y10-12). To survey about a new uniform, the SRC randomly picks 25 juniors and 35 seniors. Name the sampling method and explain why these proportions are appropriate. 2 marks
Extension — fix the bias
3.7 A market researcher samples shoppers between 10 am and 12 pm on a Tuesday outside a city café to find out what Sydney commuters think of the morning train timetable. (a) Name the method. (b) Give two reasons the sample is biased. (c) Suggest a better method. 3 marks
3.8 A student claims: "If we just ask 10,000 people instead of 100, our sample will definitely be unbiased." Using Lesson 2's misconceptions card, explain why this claim is wrong. Give a one-sentence counter-example. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (1,200-student stratified sample)
Step 1: the entire 1,200 students.
Step 2: Sample = 30 + 30 + 30 = 90 students.
Step 3: This is stratified random sampling.
Step 4: Each year group is represented in the sample in proportion to the population, so opinions of all year levels are captured.
Step 5: No — Year 10 would be under-represented (only 30/90 = 1/3 of the sample for 600/1,200 = 1/2 of the population). Sample sizes should match strata proportions.
3.1 — Names in a hat
Random sampling. Every student has an equal chance of being selected.
3.2 — Every 10th customer
Systematic sampling. A fixed rule (every kth person) is used to select the sample.
3.3 — Friends and family
Convenience sampling — and biased, because friends and family are not representative of the population.
3.4 — Every household
This is a census (data from the entire population), not a sample.
3.5 — Phone-in radio poll
Method = voluntary response sampling (a special kind of self-selected convenience sample). It is biased because only listeners with strong opinions bother to ring in, so the 4,000 callers systematically over-represent that group. Lesson 2: "sample size alone does not eliminate bias".
3.6 — Stratified by junior/senior
Method = stratified random sampling. The proportions match the population: juniors are 250/600 ≈ 41.7%, seniors 350/600 ≈ 58.3%. The sample chosen is 25/60 ≈ 41.7% juniors and 35/60 ≈ 58.3% seniors — the same proportions.
3.7 — Café market researcher
(a) Convenience sampling.
(b) Two sources of bias: (i) people sitting at a café on a Tuesday morning are unlikely to be commuters who actually use the morning train — commuters are at work; (ii) only one location is sampled, missing all commuters from other suburbs.
(c) Better method: systematic random sampling at train stations during peak morning hours (e.g. every 20th passenger boarding a CBD-bound train at multiple stations).
3.8 — "More people = unbiased" claim
The claim is wrong because bias depends on the method of selection, not the size. Lesson 2: "A convenience sample of 10,000 is still biased if it over-represents one group." Counter-example: a poll of 10,000 lunchtime gym-goers about how often Australians exercise will overstate exercise rates no matter how large 10,000 sounds.