Understand the idea.
Identify the target before collecting data
Define the population as everyone or everything the conclusion is intended to describe. The sample is the subset actually observed. A census attempts to observe the entire population, although missing responses and recording mistakes can still occur.
A parameter describes the population, while a statistic is calculated from a sample. The sample bus-travel proportion estimates the population bus-travel proportion; it is not automatically equal to it, and another random sample may give a different value.
Give members a fair opportunity to be included
A random selection uses chance rather than convenience. Its sampling frame must cover the intended population. In proportional stratified sampling, divide the population into meaningful groups and assign sample places in proportion to each group's size.
Randomly select within every stratum after allocating places. A large voluntary survey can still favour particular views or people. Evaluate coverage, the selection method and non-response before extending a conclusion beyond the people who answered.
See the method in action.
Who is the population?
A school wants to estimate how all 600 students travel to school. It randomly surveys 60 students from the enrolment list. Identify population, sample and a possible statistic.
Worked solution
- 1
Identify whom the conclusion concerns.
The target population is all 600 students enrolled at the school.
- 2
Identify the observed subset.
The sample consists of the 60 students randomly selected from the enrolment list.
- 3
Define the sample bus-travel proportion.
Here counts bus travellers among the 60 sampled students; their sample proportion estimates the bus proportion among all 600 students.
Population:600 enrolled students. Sample:60 selected students. A statistic: the proportion of the 60 who travel by bus.
Allocate a proportional sample
A college has 360 junior and 240 senior students. Plan a proportional stratified sample of 50.
Worked solution
- 1
Calculate the total population.
Both strata must be included in the denominator for proportional allocation.
- 2
Allocate the junior places.
Juniors make up three fifths of the college, so receive three fifths of the sample places.
- 3
Allocate the senior places.
Seniors receive the remaining two fifths, and the allocations sum to 50.
- 4
Select students within each stratum.
Randomly choose 30 juniors and 20 seniors from the respective complete lists; taking the first names would not be random selection.
Randomly select 30 juniors and 20 seniors.
Try it yourself.
Check your understanding
A poll asks only students who arrive early. What is one concern?
Show answer & reasoning
Arrival time may be related to travel mode or commute length.
Surveying only early arrivals excludes later arrivals. If arrival time is related to travel mode or commute length, the selected students may give a misleading picture of the whole school.
Keep practising
6 worksheet questions · 16 marks
Read the questions online
- A factory tests 40 bulbs from a day's output of 8,000. Identify population and sample.2 marks
- Explain one advantage and one limitation of a census.3 marks
- A random sample of 80 is needed from 500 members. Describe a valid selection method.3 marks
- There are 240 weekday and 160 weekend members. Allocate a proportional sample of 60.2 marks
- A website asks visitors to vote on school lunch quality. Explain two limits on a school-wide conclusion.3 marks
- A sample finds 28 of 70 students cycle. Calculate the sample proportion and explain why it is not automatically the population proportion.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A factory tests 40 bulbs from a day's output of 8,000. Identify population and sample.2 marks
Match the intended production group to the subset measured.
Population: the 8,000 bulbs; sample: the 40 tested. - Explain one advantage and one limitation of a census.3 marks
Full coverage avoids sampling variation only if observations are complete and accurate.
It aims to include everyone, but it can be costly and still suffer non-response or error. - A random sample of 80 is needed from 500 members. Describe a valid selection method.3 marks
Each member is in the frame and has an equal chance.
Do not allow repeated selections.
Number the members 1–500, then use a random generator to select 80 distinct numbers. - There are 240 weekday and 160 weekend members. Allocate a proportional sample of 60.2 marks
; the remaining 24 are weekend members.
36 weekday and 24 weekend members. - A website asks visitors to vote on school lunch quality. Explain two limits on a school-wide conclusion.3 marks
The sampling frame excludes some students.
Voluntary response can distort the sample.
Students who do not visit cannot respond; those with strong opinions may be more likely to vote. - A sample finds 28 of 70 students cycle. Calculate the sample proportion and explain why it is not automatically the population proportion.3 marks
.
The statistic estimates an unknown population quantity.
0.40, or 40%; a different random sample may give a different result.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.
Official curriculum source ↗