AP Statistics (AP Stats): Sampling Methods (SRS, Stratified, Cluster, Systematic, Convenience)

AP Statistics – Sampling Methods (SRS, Stratified, Cluster, Systematic, Convenience)

AP Statistics: Sampling Methods Study Guide

Topic: Sampling Methods (SRS, Stratified, Cluster, Systematic, Convenience)

What This Is

Sampling methods determine how we select individuals from a population to gather data. The AP exam tests your ability to identify, compare, and justify sampling methods in real-world contexts (e.g., polling voters, testing product quality, or estimating average household income). A poor sampling method can lead to biased results, making your conclusions invalid—so mastering this is critical for both FRQs and multiple-choice questions.

Real-world example: A school wants to estimate the average time students spend on homework. Instead of surveying every student (a census), they select a sample using one of the methods below. The method they choose affects whether their estimate is accurate and unbiased.

Key Terms & Formulas

• Population: The entire group of individuals we want information about (e.g., all high school students in the U.S.).
• Sample: A subset of the population used to represent the whole (e.g., 500 randomly selected high school students).
• Census: Collecting data from every individual in the population (rare due to cost/time).
• Simple Random Sample (SRS): Every possible sample of size n has an equal chance of being selected. Use randInt(lower, upper, n) on TI-84 to generate random numbers.
• Stratified Random Sample: Divide the population into homogeneous groups (strata), then take an SRS from each stratum (e.g., stratify by grade level before sampling students).
• Cluster Sample: Divide the population into heterogeneous groups (clusters), randomly select entire clusters, and survey all individuals in those clusters (e.g., randomly select 5 classrooms and survey every student in them).
• Systematic Sample: Select every kth individual from a list (e.g., survey every 10th student entering the cafeteria). k = population size / desired sample size.
• Convenience Sample: Select individuals who are easiest to reach (e.g., surveying friends). Biased and unreliable—avoid on the AP exam!
• Bias: A systematic error that favors certain outcomes. Common types:
• Undercoverage: Some groups are left out of the sampling process (e.g., phone surveys miss people without phones).
• Nonresponse: Individuals chosen for the sample don’t respond (e.g., people ignoring a survey email).
• Response Bias: Answers are influenced by question wording or interviewer behavior (e.g., "Do you agree that homework is a waste of time?").
• Random Number Generator (TI-84): MATH-PRB-randInt(lower, upper, n) to select an SRS.

Step-by-Step / Process Flow

How to answer an FRQ about sampling methods:
1. Identify the sampling method used in the scenario (SRS, stratified, cluster, systematic, or convenience).
2. Explain how the sample was selected (e.g., "The school divided students by grade and randomly selected 20 from each grade").
3. Evaluate potential biases (e.g., "Convenience sampling may overrepresent students who eat lunch early").
4. Compare to other methods (e.g., "Stratified sampling ensures representation from each grade, while cluster sampling would be easier but may miss some grades").
5. Justify why the method is (or isn’t) appropriate for the goal (e.g., "Stratified sampling is best here because grade level likely affects homework time").

Example FRQ setup: A principal wants to estimate the average number of hours students spend on homework per night. She divides students by grade (9–12), randomly selects 25 students from each grade, and surveys them. - Step 1: This is a stratified random sample (divided by grade, SRS within each grade). - Step 2: The sample was selected by first grouping students by grade, then using random numbers to pick 25 from each. - Step 3: Potential bias: If some grades have more students than others, the sample may not be proportional. - Step 4: Alternative: A cluster sample (randomly select 4 homerooms and survey all students in them) would be easier but might miss some grades. - Step 5: Stratified is better here because homework time likely varies by grade.

Common Mistakes

• Mistake: Confusing stratified and cluster sampling.
• Correction: Stratified = homogeneous groups, sample from each group. Cluster = heterogeneous groups, sample entire groups.

• Why? Stratified ensures representation from all subgroups; cluster is logistically easier but may miss some groups.

• Mistake: Calling a systematic sample an SRS.

• Correction: Systematic sampling is not random—it follows a pattern (e.g., every 10th person). Only SRS gives every possible sample an equal chance.

• Why? The AP exam penalizes mislabeling sampling methods.

• Mistake: Ignoring bias in convenience samples.

• Correction: Always state that convenience samples are biased and not generalizable to the population.

• Why? The AP exam tests whether you recognize unreliable methods.

• Mistake: Forgetting to check conditions for inference (e.g., 10% condition for SRS).

• Correction: Even if not asked, mention that the sample must be <10% of the population when sampling without replacement.

• Why? The AP rubric often awards "holistic" points for noting assumptions.

• Mistake: Using randInt incorrectly on the TI-84.

• Correction: randInt(1, 100, 5) generates 5 unique numbers between 1 and 100. Use randIntNoRep if duplicates aren’t allowed.
• Why? The exam may ask you to describe how to select an SRS, and calculator steps matter.

AP Exam Insights

• Tricky Distinction: The AP exam loves to test stratified vs. cluster sampling. Remember:
• Stratified = "some from all" (sample from every stratum).
• Cluster = "all from some" (sample entire clusters).
• Common FRQ Setup: You’ll be given a scenario (e.g., a school survey) and asked to:
• Identify the sampling method.
• Describe how to select the sample.
• Explain a potential source of bias.
• Suggest a better method.
• Calculator Pitfall: randInt can generate duplicates. If the problem says "without replacement," use randIntNoRep (or manually remove duplicates).
• Bias is Key: The AP exam always expects you to discuss bias when evaluating sampling methods. Even if the question doesn’t ask, mention it!

Quick Check Questions

1. Multiple Choice: A researcher wants to estimate the average height of trees in a forest. She divides the forest into 50 equal-sized plots, randomly selects 5 plots, and measures every tree in those plots. This is an example of:
2. (A) Simple random sample
3. (B) Stratified random sample
4. (C) Cluster sample
5. (D) Systematic sample

6. (E) Convenience sample Answer: (C) Cluster sample. The researcher samples entire plots (clusters), not individual trees.

7. FRQ Part: A school newspaper wants to survey students about their favorite lunch option. They ask the first 50 students who enter the cafeteria.

8. (a) Identify the sampling method.
9. (b) Explain one potential source of bias. Answer:
10. (a) Convenience sample.

11. (b) Response bias: Early lunch-goers may have different preferences than those who eat later.

12. Multiple Choice: Which of the following is not a source of bias in sampling?

13. (A) Undercoverage
14. (B) Nonresponse
15. (C) Using a random number generator
16. (D) Leading question wording
17. (E) Convenience sampling Answer: (C) Using a random number generator. Random selection reduces bias; the other options introduce it.

Last-Minute Cram Sheet

1. SRS: Every sample of size n has equal chance. Use randInt.
2. Stratified: "Some from all" (homogeneous groups). Reduces variability.
3. Cluster: "All from some" (heterogeneous groups). Easier but riskier.
4. Systematic: Every kth individual. Not random!
5. Convenience: Biased—never use on the AP exam.
6. Bias types: Undercoverage, nonresponse, response bias.
7. 10% condition: Sample must be <10% of population for independence.
8. TI-84: randInt(lower, upper, n) for SRS.
9. Stratified-Cluster! Stratified = homogeneous groups; cluster = heterogeneous.
10. Always mention bias when evaluating sampling methods.