Start with What Was Selected
In Random Condition for Proportion Inference, you learned to ask what was randomized and what population a conclusion targets. This tutorial adds a closer look at four sampling methods that may appear in a study description: simple random sampling, stratified sampling, cluster sampling, and voluntary response. The key is to follow the selection process step by step.
First identify the population and the units the researcher could select. Then ask whether the researcher selected individual units or groups, whether selection used chance, and whether people chose for themselves to respond. These details—not the size of the sample and not the mere presence of the word “random”—determine how to classify the method.
Four Sampling Methods to Recognize
A simple random sample (SRS) of size \(n\) is selected so that every possible group of \(n\) individuals from the population has an equal chance of being the sample. In a description, look for a chance method used to select individuals from a complete or appropriate list, such as a random-number generator. Randomly selecting individuals is different from asking people to volunteer or choosing whoever is easiest to contact.
In stratified sampling, the researcher divides the population into groups called strata and then selects a random sample from every stratum. Strata are often formed using a characteristic relevant to the study, such as grade level or region. The important clue is that the researcher samples from each stratum. Individuals, rather than whole strata, are sampled within those groups.
In cluster sampling, the population is divided into natural groups called clusters. The researcher randomly selects some clusters and collects data from all individuals in the chosen clusters, or sometimes from a random sample of individuals within them. The defining clue is that some clusters are selected and others are not. If every cluster is represented by sampling individuals within each one, that is stratified sampling, not cluster sampling.
In a voluntary-response sample, people decide for themselves whether to participate, often after an open invitation. A poll posted online that anyone can answer is a familiar example. This is a form of self-selection: people who feel strongly or have easy access to the invitation may be more likely to respond. It is not random sampling, even if a large number of people respond.
| Method | What is selected? | Does chance guide selection? |
|---|---|---|
| Simple random sample | Individuals from the population | Yes; every possible sample of the stated size has an equal chance |
| Stratified sample | Individuals sampled within every stratum | Yes; individuals are randomly selected within each stratum |
| Cluster sample | Some groups, often followed by all individuals in those groups | Yes; clusters are randomly selected |
| Voluntary response | People who choose to answer an invitation | No; individuals select themselves into the sample |
The names of the groups alone do not settle the classification. For example, a researcher might divide a school into classrooms and then select students at random from every classroom. That is stratified sampling. If the researcher randomly picks a few classrooms and surveys everyone in those classrooms, that is cluster sampling.
A Reliable Classification Routine
Use the following routine when a description is unfamiliar. Do not start by asking whether the sample “looks representative.” First determine exactly how the sample was formed. Then make a separate judgment about whether the design supports the Random condition for the intended inference.
Identify who the conclusion is about and what the researcher could select: people, households, classrooms, clinics, or other units.
If people choose whether to answer an open invitation, classify the sample as voluntary response. A chance-based invitation does not make the responses random if individuals choose whether to participate.
Randomly select individuals from the population for an SRS. Randomly sample individuals within every group for a stratified sample. Randomly select some groups for a cluster sample.
Decide whether chance-based sampling supports inference to the population represented by the sampling process. State the target population precisely and note important limits of the sampling frame.
A random sampling method does not automatically make every one-proportion procedure appropriate. As emphasized in Why Inference Procedures Need Conditions, the conditions serve different purposes. The Random condition concerns how the sample was obtained. The 10% condition and the Large Counts condition are separate checks; the next tutorial focuses on the 10% condition.
Also keep the target population in view. An SRS from a list of current library members can support inference to the members represented by that list, not automatically to every resident of the area. A chance-based method cannot include people who are missing from its sampling frame.
Worked Examples
Worked Example: Selecting Individuals Directly
A fictional recreation department has a complete list of 1,800 people enrolled in its summer programs. A computer randomly selects 90 individuals from the list. The department asks whether each selected person plans to attend a program event. Of the 90 selected people, 54 say yes. The department wants to estimate the proportion of enrolled people on the list who plan to attend.
State. The parameter of interest is the proportion of enrolled people represented by the list who plan to attend the event. The issue is whether the described sample supports the Random condition.
Plan. Classify the method by identifying what was selected and whether selection used chance. Then check whether the intended population matches the sampling frame.
Do. The computer selected individuals directly and did so randomly. This is an SRS, not a stratified sample because the description gives no separate sampling within groups, and not a cluster sample because no groups were selected. The sample proportion is \(\hat{p}=54/90=0.60\), or 60%. The list is stated to be complete for enrolled people, so the selection process matches the target named in the question.
Conclude. The SRS supports the Random condition for making an inference about the proportion of people enrolled in these programs who plan to attend, subject to the other conditions for the chosen procedure. It does not support a claim about all people in the community, because the sample was drawn from program enrollees rather than the entire community.
Worked Example: Sampling Within Every Grade
A fictional high school has 600 students: 180 in grade 9, 150 in grade 10, 140 in grade 11, and 130 in grade 12. A researcher randomly selects 20 students from each grade and asks whether they have used the school library this month. Among the 80 selected students, 52 say yes. The researcher is interested in the proportion of all students at the school who used the library this month.
Identify the design. The researcher divided the population into grade-level strata and randomly selected students from every grade. This is stratified sampling. The researcher did not select entire grades or a few classrooms as groups, so this is not cluster sampling.
Decide whether the Random condition is met. The selection within each grade used chance, and students from every grade were eligible to be selected. The target is all students at this school, which matches the four grades in the described population. The stratified random sample therefore supports the Random condition for inference to the school’s students, assuming the selection and data collection were carried out as described.
Report the observed proportion carefully. The overall sample proportion is \(\hat{p}=52/80=0.65\). It describes the 80 sampled students, with each grade contributing equally to the pooled proportion despite the grades having different population sizes. To estimate the whole-school proportion, weight each grade’s sample proportion by that grade’s share of the 600 students; the total of 52 students reporting yes does not provide the grade-specific results needed for that estimate. The random design provides a basis for inference about all students, but the design alone does not establish the other conditions for a one-proportion procedure. A standard one-proportion procedure using the pooled proportion is not justified by the design alone; an analysis should account for the stratified allocation.
Worked Example: Choosing Classrooms at Random
A fictional district has 48 elementary-school classrooms, with 24 students in each classroom. A researcher randomly selects 6 classrooms and surveys every student in those classrooms about whether they bring a reusable water bottle to school. In total, 144 students are surveyed, and 81 report bringing one. The target is all 1,152 students in the district’s elementary classrooms.
Identify the design. The researcher randomly selected some natural groups—classrooms—and surveyed everyone in the selected groups. This is cluster sampling. It is not an SRS of 144 students, because students were selected as members of chosen classrooms rather than as individuals from the full district list. It is not stratified sampling, because the researcher did not take a random sample of students from every classroom.
Check the Random condition. The classrooms were selected using chance, and the target population is the students in those classrooms across the district. The described method is a random cluster sample and supports the Random condition for inference about that target, provided the classrooms form an appropriate frame and the selection process was followed.
Summarize the result and its limits. The observed proportion is \(\hat{p}=81/144=0.5625\), or 56.25%. This is the proportion among the students in the six selected classrooms. The cluster design should be named rather than mislabeled as an SRS. The Random condition is only one part of the procedure’s justification; do not treat this classification as a substitute for checking other applicable conditions or considering whether the sampling frame covers the intended population.
Worked Example: An Open Online Poll
A fictional town posts a poll on its public website asking, “Should the town add a weekend bus route?” Anyone visiting the page can choose to submit an answer. By the closing date, 1,250 people have responded, and 875 support the route. Town staff want to estimate the proportion of all town residents who support it.
Identify the design. Respondents chose whether to answer an open invitation. This is a voluntary-response sample, not an SRS, stratified sample, or cluster sample. The large response count does not change how the participants entered the sample.
Calculate what the responses describe. Among the people who responded, the proportion supporting the route is \(\hat{p}=875/1{,}250=0.70\), or 70%. This accurately summarizes the responses collected.
Decide whether the Random condition is met for the intended inference. The town did not randomly select residents; people selected themselves by choosing to answer. Residents who care strongly about bus service or who frequently visit the website may be more likely to respond. Therefore, the poll does not support the Random condition for a one-proportion inference about all town residents. The 70% result should be described as the proportion among respondents, not as a statistically justified estimate for all residents.
Common Mistakes and AP Exam Tip
A frequent error is to identify a method from just one detail, such as the fact that the population was divided into groups. Instead, follow what happens after the groups are formed: are individuals sampled from every group, or are some entire groups selected? That distinction separates stratified from cluster sampling.
- Calling any grouped sample “cluster sampling.” If individuals are randomly sampled from every group, the design is stratified. Cluster sampling selects some groups, often surveying everyone in those groups.
- Calling a large sample random. The number of responses does not tell you how they were obtained. An open poll remains voluntary response even with thousands of replies.
- Confusing invitation with selection. A researcher may invite many people, but if people decide for themselves whether to respond, the resulting sample is self-selected.
- Stopping at “random sample.” State the type of random sample and the population it represents. A random sample from a limited list supports inference only to the population that list adequately covers.
- Claiming the Random condition guarantees a sound conclusion. It is one condition, not the whole analysis. Check the other requirements for the intended procedure and consider limitations such as nonresponse or an incomplete sampling frame.
Key Takeaway
Classify a sampling method by tracking who or what was selected, whether chance guided selection, and whether individuals chose to respond. An SRS selects individuals directly; a stratified sample selects individuals within every stratum; a cluster sample selects some groups; and a voluntary-response sample consists of people who choose to answer.
Check Your Understanding
Classify each sampling method and decide whether it supports the Random condition for the stated target.
- A clinic randomly selects 75 patient records from a complete list of its patients to ask whether patients received a particular vaccine. What sampling method is used, and what population might the sample represent?
- A researcher randomly selects 12 students from each of five randomly unselected grade levels at a school. Is this stratified or cluster sampling? Explain which detail determines the classification.
- A researcher randomly selects four school buses from a district and surveys every student riding those buses. What is the sampling method?
- A neighborhood group posts a survey link and uses the answers submitted by residents who choose to respond. Does the result meet the Random condition for inference to all neighborhood residents? Why or why not?
- A company randomly samples employees from a list containing only its head-office staff, but wants to make a claim about employees at all locations. What limitation should be mentioned?