Understand the idea.
Begin with the target population
Define the population by asking whom the final claim concerns. Then identify the variable measured on each person or item. The sampling frame is the practical list used for selection; compare it with the population before trusting any calculation.
In a simple random sample of fixed size, every possible sample of that size has equal selection probability. Random selection protects against deliberate selection preferences, but cannot include people missing from the frame. A voluntary poll uses a different selection mechanism.
Choose a design and inspect its weaknesses
Stratify when relevant groups need representation: divide the population into non-overlapping groups, allocate sample places, then select randomly within each. For proportional allocation, use . Systematic sampling instead uses a random start and fixed interval; check for repeating patterns.
Separate sampling variation from bias. Variation changes between random samples; bias systematically favours particular results. An incomplete frame, selective non-response or leading wording can cause bias. Increasing the sample count reduces random variation, but does not remove those mechanisms.
See the method in action.
Allocate a stratified sample
There are 360 junior and 240 senior students. Allocate a proportional random sample of 50.
Worked solution
- 1
Find the total population.
The allocation must use each group as a fraction of the whole school.
- 2
Allocate the junior places.
Juniors form the same fraction of the sample as of the population.
- 3
Allocate the senior places and check the total.
The two allocations account for all sample places.
- 4
Select within each group.
Randomly choose 30 distinct junior labels and 20 distinct senior labels from complete lists; volunteers would not implement the stated design.
30 juniors; 20 seniors.
Examine the frame
A voluntary cycling-club poll has 8,000 replies. Can it establish the cycling rate of all local adults?
Worked solution
- 1
Compare the audience with the target population.
The target is all local adults, whereas the club poll reaches a cycling-club audience and can exclude other adults.
- 2
Inspect the decision to reply.
People interested in cycling may be more likely to volunteer an answer, so respondents need not resemble the target population.
- 3
Evaluate the large reply count.
A large count within this selection mechanism does not remove frame or response bias.
- 4
State the limit on the conclusion.
The poll cannot establish the cycling rate of all local adults without addressing selection and response bias.
Not without addressing selection and response bias.
Try it yourself.
Check your understanding
A random sample uses a list excluding part-time students. Is it adequate for all students?
Show answer & reasoning
Not without addressing the excluded group.
Part-time students have no chance of selection from this list. Random selection among the listed students does not repair that coverage gap, so the frame must be addressed before generalising to all students.
Keep practising
6 worksheet questions · 12 marks
Read the questions online
- Define the population and variable in a study of weekly bus trips by Year 12 students at one school.2 marks
- A college has 420 day and 180 evening students. Allocate a proportional sample of 40.2 marks
- Describe a simple random sample of 15 from a complete list of 200 items.2 marks
- A list repeats machines A, B, C, D. Why might every fourth item be unsuitable?2 marks
- Households are contacted only once at 2 pm on a weekday. Identify a likely bias.2 marks
- Rewrite “Should our excellent new timetable remain?” neutrally.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Define the population and variable in a study of weekly bus trips by Year 12 students at one school.2 marks
Match the group to the claim.
Identify the numerical measurement for each selected student.
All Year 12 students at that school; each student’s weekly trip count. - A college has 420 day and 180 evening students. Allocate a proportional sample of 40.2 marks
Total ; and .
28 day; 12 evening. - Describe a simple random sample of 15 from a complete list of 200 items.2 marks
Every fixed-size selection must have the same chance.
Redraw duplicate labels rather than substituting a convenient item.
Number items 1–200 and choose 15 distinct random labels. - A list repeats machines A, B, C, D. Why might every fourth item be unsuitable?2 marks
The interval matches the list’s period.
Randomise the order or deliberately sample across machine types.
The sampling interval may repeatedly select one machine type. - Households are contacted only once at 2 pm on a weekday. Identify a likely bias.2 marks
Availability can relate to the measured characteristic.
Use repeat contacts at varied times and report residual non-response.
Non-response associated with being away at work or study. - Rewrite “Should our excellent new timetable remain?” neutrally.2 marks
Remove the value judgement “excellent”.
Offer balanced choices without suggesting a preferred answer.
Should the new timetable remain, be changed, or be replaced?
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 Mathematical Methods: sampling distributions and confidence intervals for means and proportions. Mean tasks distinguish known population spread from sufficiently large-sample estimated-spread normal intervals. Repeated-interval computational simulation, approximation conditions, design and interpretation are included.
Official curriculum source ↗