Ways to choose a sample

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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Students will be able to

• I can describe simple random, systematic, stratified and convenience sampling.
• I can work out a systematic step and a stratified share.
• I can choose a sampling method and say why it is fair.

Timing

Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need census, sample and bias (last two lessons).

The one idea: random methods (simple random, systematic from a random start, stratified) give every member a chance; convenience and quota let a person choose, so they can be biased. The common mistake is calling a hand-picked sample random (Ava's mistake).

The board lists the school by year group, so a systematic sample spreads through all years; a stratified sample guarantees 5 per year; a convenience sample is the front rows (Year 7).

Quota sampling (typical width in the curriculum's progression) is in the lesson's table and one Modify question. Wider: random number generators (=RANDBETWEEN(1,200)) to choose a real sample from a class list.

Running the lesson

1. Predict: ask what could make a random sample uneven.
2. Run: ask how many came from each year with each method.
3. Investigate: ask Ava who chose the students.
4. Modify: ask when a systematic sample goes wrong.
5. Make: ask where each method's results sit.

The hard part (on purpose)

Random isn't the same as 'picked without thinking'. Ask: did every student have the same chance, and who decided?

When students get stuck

Answers to the checks

1. a fair share: You pick 20 of the 200 students completely at random. Will you get exactly 5 from each year group?
   Not always. A random sample is fair, but by chance it can have 3 Year 7s and 8 Year 10s. To promise 5 from each, choose 5 at random within each year.
2. four methods: Take a simple random sample and read how many came from each year (you can take a few). Then take a stratified sample. Which promises 5 from each year?
   
   Stratified sampling chooses at random within each year group, so each gets its fair share: 5 of 20.
3. Every kth: A systematic sample of 40 is taken from a list of 200. Every how many students is chosen?
   5
   200 ÷ 40 = 5: start at a random student among the first 5, then take every 5th.
4. Find the mistake: Ava chooses a sample. Click the line where her method stops being random.
   line 2
   Line 2: Ava chose the students herself, so who was around at lunch, and whom she noticed, decides the sample. Random means every student has the same chance, such as numbers from a random number generator.
5. Why stratify: Year 7s sleep more than Year 10s. Why is a stratified sample (by year) a good choice for a sleep survey?
   Each year group gets its fair share, so no year is over- or under-represented
   When groups differ on what you measure, stratifying stops chance from tipping the sample towards one group.
6. Systematic: Take a systematic sample of 20. Which students does it pick?
   
   Every 10th student from a random start (200 ÷ 20 = 10), spread evenly through all four years: 5 from each.
7. In proportion: A school has 300 Year 7s and 200 Year 8s. A stratified sample of 50 is taken in proportion. How many Year 7s are in it?
   30
   300 ÷ 500 = 3/5, and 3/5 of 50 = 30. Then 20 Year 8s.
8. A patterned list: A list of 200 students alternates boy, girl, boy, girl. Why is a systematic sample with every 2nd student a bad idea?
   It would pick all boys or all girls
   When the list has a pattern that matches the step, a systematic sample is biased.
9. Quota: An interviewer is told to ask 10 people from each age group, choosing whoever she likes in a shopping centre. What kind of sample is this?
   A quota sample: fixed shares, but not chosen at random
   Quota samples fill shares like stratified ones, but the collector chooses who, so bias can creep in.
10. Names in a hat: Each student's name is put in a hat and 30 are drawn. What method is this?
    Simple random
    Every student has the same chance of being drawn.
11. Every 50th: A factory tests every 50th phone off the production line. What method is this?
    Systematic
    A fixed step along a list (or line) is systematic.
12. Seniors: A club has 80 junior and 40 senior members. A stratified sample of 15 is taken in proportion. How many seniors?
    5
    Seniors are 40 of 120 = 1/3, and 1/3 of 15 = 5.
13. Review: bias: Review: a survey of people at a vegan festival asks how often they eat meat. What is the main problem?
    Sampling bias: festival-goers eat less meat than most people
    Who was asked pushes the result down.
14. pick the best method: Take 10 samples of 20 with a fair method of your choice (not convenience). Do its results sit around the population's mean?
    
    Fair methods give results scattered around the population's mean; convenience samples sit to one side.

Solutions

One possible solution for each task. Other correct programs are fine.

use the methods › Systematic

Open task: any program that does what the step asks.

use the methods › In proportion

Open task: any program that does what the step asks.

use the methods › A patterned list

Open task: any program that does what the step asks.

use the methods › Quota

Open task: any program that does what the step asks.

pick the best method

Open task: any program that does what the step asks.

Stretch and support

Stretch: Take a real stratified sample of 12 from your class list (by a group of your choice) using a random number generator. Write each step.

Support: The four methods table as a card, with the board's picture for each.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Ways to choose a sample".

The idea: Comparing sampling methods: I can Students compare simple random, systematic, stratified and convenience sampling methods and evaluate the reliability of each.

What understanding looks like:
- Describes at least three different sampling methods.
- Explains the reliability advantage of random over convenience sampling.
- Selects and justifies an appropriate sampling method for a given context.

Mistakes students often make here:
- A student thinks random means 'whoever I happen to pick'.
- A student divides the wrong way for the systematic step.
- A student gives every stratum the same number when the groups differ in size.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Ways to choose a sample".

Exit ticket

A school has 400 Year 9 and 200 Year 10 students. Describe how to take a stratified sample of 30.

In proportion: 20 Year 9s and 10 Year 10s, each chosen at random from its year's list.

What each step is for

• a fair share: Committing to which method guarantees each year group a share shows whether they see what random choice can and can't promise.
• four methods: Taking each kind of sample and reading how many came from each year shows how the methods differ.
• how each works: Find k for systematic sampling; picking people yourself isn't random even if you try to be fair (Ava); stratify when groups differ.
• use the methods: Running systematic and stratified samples and working out shares makes the methods concrete.
• choosing a method: Naming and choosing methods in new contexts, mixed; working out stratified shares; bias is the review.
• pick the best method: Comparing methods on the same population by their results shows which are fair and which are biased.

Where it leads

This starts AC9M8ST02.

Where it leads: describing a sample's distribution (next), sampling variation, and investigations.