AP Statistics Unit 1: Random sampling methods

What the exam asks

Under Statistical Practice 2 (Collect Data), the exam tests five core tasks:

Identify/classify a sampling method (simple random, stratified random, cluster random, or systematic random) from a scenario.
Distinguish sampling with replacement from sampling without replacement.
Describe how to implement a simple random sample (SRS) using a random number generator (RNG) or slips of paper.
Explain how strata and clusters are formed and sampled.
Justify why one sampling method is better than another for a specific investigative question and population.

Without replacement: Once an individual is selected, they cannot be selected again. Almost all real-world surveys sample without replacement so the same person is not interviewed twice.
With replacement: A selected individual is placed back into the pool and can be chosen again. This keeps selection probabilities constant and independent across draws.

Method	How the population is grouped	How the sample is selected	Key advantage
Simple random sample (SRS)	No grouping; uses a single list (sampling frame) of the entire population of size $N$ .	Every possible group of $n$ individuals has an equal chance of being chosen.	Unbiased baseline method; requires no prior knowledge of population subgroups.
Stratified random sample	Split into strata: groups that are homogeneous (similar within each group, different between groups) on a variable tied to the response.	Take a separate SRS from every stratum and combine them.	Guarantees representation of every group and reduces variability in estimates.
Cluster random sample	Split into clusters: natural groups (often geographic or administrative) that are internally heterogeneous and each mirror the population.	Take an SRS of clusters and measure every individual (all) in the chosen clusters.	Saves time and money when a full list of individuals is unavailable or subjects are spread out.
Systematic random sample	Ordered in a line, sequence, or list.	Use a random method to pick a starting point among the first $k$ individuals, then select every $k\text{th}$ individual thereafter.	Easy to carry out in the field (e.g., exit polls or factory assembly lines) without a full advance list.