The idea, step by step
Begin with the target population
The population is the complete group addressed by a claim. The sampling frame is the list or practical source used to select units. Random selection from an incomplete frame can still miss an important group. State the population, variable and selection method before calculating anything.
In a simple random sample of fixed size, every possible sample of that size has the same chance of selection. A large voluntary poll is not random merely because many people reply.
Choose a design and inspect its weaknesses
Stratified sampling divides a population into relevant groups and randomly samples within each group. Proportional allocation represents group sizes. Systematic sampling uses a random start and fixed interval, but periodic patterns can make it unrepresentative.
Bias is systematic distortion rather than ordinary sampling variation. Non-response may leave respondents unlike the selected sample. Leading wording can alter answers. More observations do not repair the mechanism producing bias.
A common trap
Large samples must be representative.
Instead: Representativeness depends on selection and response, not count alone.
See the thinking
Allocate a stratified sample
There are 360 junior and 240 senior students. Allocate a proportional random sample of 50.
- Total . Allocate and .
- Randomly choose distinct labels within each stratum; do not replace the draw with volunteers.
Examine the frame
A voluntary cycling-club poll has 8,000 replies. Can it establish the cycling rate of all local adults?
- The club audience is not a complete frame of local adults.
- Interested cyclists may be more likely to respond. A larger number of similar respondents does not fix that problem.
Check your understanding
A random sample uses a list excluding part-time students. Is it adequate for all students?
Show answer
Not without addressing the excluded group.
Random selection cannot repair an incomplete frame.
Your turn
6 questions · 12 marks · Work at your own pace.
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 ↗