Statistical investigation and sampling
Identify and classify sampling methods, evaluate survey questions for flaws, and assess whether a sample is representative of the population.
Worked examples
Identifying a sampling method
Straightforward
Problem
A Year 11 class of 30 students wants to select 6 students to represent them at a school council meeting. The teacher writes each student's name on a separate slip of paper, places all slips in a bag, shakes it, and draws out 6 slips without looking. Identify the sampling method used and explain why it is appropriate.
1
Determine whether every student has an equal chance of being selected.
All 30 names are in the bag. Each draw is made without looking, so every student has an equal chance of on the first draw.
2
Identify the sampling method.
Selection is entirely by chance with no fixed pattern and no grouping of students — this is random sampling.
3
Explain why the method is appropriate.
Every student could be chosen, so no individual or group is favoured. The sample is unlikely to be systematically biased.
Answer
The sampling method is random sampling. It is appropriate because every student has an equal chance of being selected, which reduces bias in choosing representatives.
Identifying and improving a flawed survey question
Moderate
Problem
A student writes this survey question to investigate screen time: "Do you spend too much time on your phone and does it stop you from sleeping?" Identify two flaws in this question and suggest a specific improvement for each flaw.
1
Check whether the question is neutral or assumes a particular answer.
The phrase "too much" assumes the respondent has a problem with phone use. This steers respondents towards agreeing, making it a leading question.
2
Suggest an improvement for the leading language.
Replace "too much" with a neutral phrase. For example: "On average, how many hours per day do you spend on your phone? (Less than 2 / 2 to 4 / More than 4)"
3
Check whether the question asks about more than one topic.
The question asks about phone usage and sleep in the same item. A respondent who uses their phone a lot but sleeps well cannot answer truthfully. This is a double-barrelled question.
4
Suggest an improvement for the double-barrelled structure.
Split into two separate questions: (1) "On average, how many hours per day do you spend on your phone?" and (2) "Does using your phone at night affect how well you sleep? (Yes / No / Sometimes)"
Answer
The question has two flaws. First, it is leading — "too much" assumes excessive use. Second, it is double-barrelled — it asks about phone use and sleep in one item. Each flaw is addressed by rewriting as a separate, neutral question.
Evaluating representativeness and identifying an ethical concern
Challenging
Problem
A researcher wants to study healthy eating habits across a city. She surveys 200 people attending a farmers market on a Saturday morning. Explain whether this sample is likely to be representative of the city's population, and identify one ethical concern with the investigation.
1
Consider the characteristics of people who attend a farmers market on a Saturday morning.
Farmers market attendees tend to be people who value fresh or locally produced food and have free time on Saturday mornings. This excludes shift workers, people with lower disposable incomes who shop elsewhere, and those who are less focused on healthy eating.
2
Conclude whether the sample represents the broader city population.
The sample is drawn from a self-selected group with above-average interest in healthy food. It overrepresents health-conscious consumers and is unlikely to reflect the full range of eating habits across the city.
3
Identify one ethical concern.
Participants approached in a public setting may feel social pressure to respond. They may not realise they are free to decline, which means consent may not be fully informed or voluntary.
Answer
The sample is unlikely to be representative — farmers market shoppers tend to have stronger interest in healthy eating than the general population, so the data will overestimate healthy eating habits city-wide. An ethical concern is that participants approached in public may not feel free to decline, raising questions about whether their consent is genuinely voluntary.
Practise
Q1·Straightforward
A school principal wants to survey 40 students from a school of 800. She places all student names in a box, mixes them thoroughly, and draws out 40 names. What type of sampling method is this?
Explanation
Every student's name is in the box and each has an equal chance of being drawn. This is random sampling — selection is determined entirely by chance with no pattern or grouping.
Q2·Straightforward
A market researcher stands at a shopping centre entrance and surveys every 10th person who enters. What type of sampling method is this?
Explanation
Selecting every 10th person follows a fixed, regular interval — this is systematic sampling. The starting point may be random, but after that, every 10th person is chosen according to a set rule.
Q3·Straightforward
A television news channel posts a poll on its website asking viewers: "Should public transport be free?" and invites them to vote. What type of sampling method does this represent?
Explanation
Only viewers who choose to visit the website and click the poll will respond. Participants decide for themselves whether to take part — this is self-selected sampling, also called voluntary response sampling.
Q4·Straightforward
A hospital employs 300 nurses and 100 doctors. A researcher randomly selects 30 nurses and 10 doctors to participate in a workplace satisfaction survey. What type of sampling method is this?
Explanation
The population is first divided into two groups (strata) — nurses and doctors — and then a proportional sample is drawn from each group. This is stratified sampling. The sampling fraction is from each stratum.
Q5·Moderate
A survey question reads: "Don't you agree that social media is harmful to teenagers?" What is the main flaw in this question?
Explanation
The phrase "Don't you agree" steers respondents towards answering yes, which biases the results. A neutral version would be: "Do you think social media is harmful to teenagers? Yes / No / Unsure."
Q6·Moderate
A questionnaire includes the question: "Do you eat a healthy diet and exercise regularly?" What is the main flaw in this question?
Explanation
This is a double-barrelled question — it asks about diet and exercise in a single item. A respondent who eats well but does not exercise regularly cannot answer truthfully with a single yes or no. Each topic should be a separate question.
Q7·Moderate
A researcher emails a survey to 500 employees but only 60 (12%) respond. What is the main concern with using these results to represent all employees?
Explanation
When a large proportion of the sample does not respond, the results may reflect only those with strong opinions or particular characteristics. This is non-response bias. For example, dissatisfied employees may be more likely to complete the survey than satisfied ones, skewing the results.
Q8·Moderate
A researcher surveys 30 students sitting in the school library at lunchtime to represent the views of all 900 students in the school. Which specific improvement would best address the main flaw in this approach?
Explanation
The main flaw is convenience sampling — only library users at lunchtime are surveyed, which may not reflect the views of all 900 students. Randomly selecting from the full school roll gives every student an equal chance of being chosen, addressing the bias. Doubling the library sample or changing the time still restricts the pool to library users.
Q9·Challenging
A study of smartphone usage recruits participants by placing an advertisement on a popular social media platform. Which statement best explains why this sample may not represent all Australian adults?
Explanation
Adults who do not use the platform — including those without internet access, older Australians, or those who have opted out of social media — cannot see the advertisement and are therefore excluded. Because the study is about smartphone usage, excluding people less engaged with technology is a significant source of bias and makes the sample unrepresentative.
Q10·Challenging
A fitness company surveys 1 000 of its gym members to draw conclusions about the exercise habits of all Australians. Why is this sample unlikely to be representative of the general population?
Explanation
Gym members have self-selected into a physically active lifestyle, so they are not typical of all Australians. Using this sample to draw conclusions about general exercise habits introduces selection bias — the sample overrepresents active people. A representative sample would need to include people across a wide range of activity levels.
Q11·Challenging
A school collects student responses to a mental health survey. Each response is linked to the student's name, and the results are later shared with prospective employers without the students' knowledge. What is the most significant concern with this investigation?
Explanation
Linking responses to names means the data is not anonymous. Sharing identifiable health information with employers without students' knowledge or consent violates privacy. Participants should be informed of how their data will be used before the investigation begins, and sensitive personal data requires explicit consent before sharing with third parties.
Q12·Challenging
A researcher films shoppers in a supermarket without their knowledge to study purchasing behaviour. Which ethical concern is most significant?
Explanation
Conducting research without participants' knowledge denies them the opportunity to give informed consent — a fundamental requirement of ethical research. Even if the behaviour observed is in a public place, recording individuals for a research study without disclosure is an ethical breach. Ethical approval and participant consent are required before data collection begins.