AP Statistics Collecting Data — Worked Answer Explanations

Unit 3 · 12 questions explained

Below is a complete answer key for our AP Statistics Collecting Data practice questions. For each question you'll find the correct choice, a full written explanation of how to get there, and — for every wrong answer — a short note on exactly why it's tempting and where it goes wrong. Reading these straight through is one of the fastest ways to find the gaps in a unit before exam day.

Prefer to test yourself first? Take the timed Collecting Data practice test and come back here to review, or head back to the Collecting Data unit overview.

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Question 1 · Easy
A school wants to estimate the proportion of students who bike to school. They survey every 10th student on the enrollment list. What type of sampling method is this?
- A
  Simple random sample (SRS)
  Why not A: An SRS requires every possible group of $n$ students to have an equal chance of selection; systematic sampling does not guarantee this.
- B
  Systematic random sampleCorrect
- C
  Stratified random sample
  Why not C: Stratified sampling divides the population into groups and randomly samples from each group.
- D
  Cluster sample
  Why not D: Cluster sampling randomly selects intact groups (e.g., homerooms), not every k-th individual.
Explanation
Systematic random sampling involves selecting every $k$ -th individual from a list after a random start. Here, every 10th student is selected. This differs from an SRS because not every subset of $n$ students has an equal chance of being selected.

Key takeaway
Systematic sampling selects every k-th unit; it is not the same as an SRS because certain combinations of units can never be chosen.
Question 2 · Easy
A researcher wants to survey voters about a proposed tax increase. She mails surveys to all registered voters, but only 8% respond. Which bias is most likely present?
- A
  Undercoverage bias
  Why not A: Undercoverage occurs when part of the population cannot be included; here the mailing list may cover all voters.
- B
  Response bias
  Why not B: Response bias involves inaccurate answers due to question wording or social pressure, not who responds.
- C
  Nonresponse biasCorrect
- D
  Voluntary response bias
  Why not D: Voluntary response involves a self-selected group who choose to respond to an open call; here surveys were mailed to a defined list.
Explanation
Nonresponse bias occurs when those who do not respond differ systematically from those who do. A response rate of only 8% is very low; if people with strong opinions (pro or con) are more likely to respond, the sample will not represent all registered voters.

Key takeaway
Nonresponse bias occurs when non-responders differ from responders in ways relevant to the survey question.
Question 3 · Easy
An experiment randomly assigns 40 patients to receive a new drug or a placebo. Neither the patients nor the doctors evaluating outcomes know which treatment each patient received. What design feature does this describe?
- A
  Replication
  Why not A: Replication means using enough subjects so results are reliable, not concealing treatment assignments.
- B
  Blocking
  Why not B: Blocking groups experimental units with similar characteristics before random assignment.
- C
  Double-blind designCorrect
- D
  Matched pairs design
  Why not D: Matched pairs pairs subjects with similar characteristics; it is not about knowledge of treatment.
Explanation
A double-blind design means neither the subjects nor the evaluators know who received which treatment. This prevents both the placebo effect (subjects believing they are treated) and evaluator bias (doctors unconsciously treating or scoring groups differently).

Key takeaway
Double-blind: neither subjects nor evaluators know treatment assignment, controlling for placebo and evaluator bias.
Question 4 · Easy
Researchers studying the effect of sleep on memory randomly assign 30 college students to either 6 hours or 8 hours of sleep, then test their recall the next morning. What is the explanatory variable?
- A
  Memory recall score
  Why not A: Memory recall is the outcome (response variable), not the factor being manipulated.
- B
  Hours of sleepCorrect
- C
  The college students
  Why not C: The students are the experimental units, not a variable.
- D
  Whether the study was blinded
  Why not D: Blinding is a design feature, not the explanatory variable.
Explanation
The explanatory variable (independent variable) is the factor the researchers manipulate — here, the number of hours of sleep (6 vs. 8). The response variable is memory recall, the outcome being measured.

Key takeaway
The explanatory variable is the one deliberately changed or assigned; the response variable is the outcome measured.
Question 5 · Easy
A study finds that towns with more fast-food restaurants have higher rates of obesity. A researcher concludes that fast food causes obesity. Which of the following is the strongest critique of this conclusion?
- A
  The sample size is too small to draw conclusions.
  Why not A: The number of towns is not specified as small; the primary issue is the study design, not sample size.
- B
  This is an observational study; confounding variables (such as income level or physical activity norms) may explain the association.Correct
- C
  Correlation between these variables is always spurious.
  Why not C: Correlations are not always spurious; the issue is whether this one supports a causal claim.
- D
  A randomized experiment should have been used to compute the correlation.
  Why not D: Randomized experiments establish causation; computing correlations is appropriate for observational data but cannot establish causation.
Explanation
This is an observational study. Without random assignment, causal conclusions cannot be drawn. Confounding variables such as income, urban density, or local exercise culture could be associated with both fast-food presence and obesity rates, creating a spurious association.

Key takeaway
Observational studies cannot establish causation; confounding variables are always a threat.
Question 6 · Easy
A stratified random sample is taken to estimate average income in a city. The population is divided into three strata by neighborhood. Why might stratification produce a more precise estimate than a simple random sample of the same size?
- A
  Stratification eliminates all bias in the sample.
  Why not A: Stratification reduces sampling variability but does not eliminate all potential biases.
- B
  Stratification ensures representation from every neighborhood, reducing variability between strata from contaminating the estimate.Correct
- C
  Stratification allows researchers to use larger sample sizes.
  Why not C: The question specifies the same sample size; stratification improves precision, not sample size.
- D
  Stratification is only useful when all strata have equal population sizes.
  Why not D: Stratification works with unequal stratum sizes; proportional allocation adjusts for different stratum sizes.
Explanation
Stratification forces the sample to include units from every stratum. If incomes vary greatly between neighborhoods, a simple random sample might, by chance, over-sample high-income neighborhoods. Stratification removes this source of variability, yielding more precise estimates when strata are internally homogeneous.

Key takeaway
Stratify when strata are internally homogeneous but differ from each other; this reduces sampling variability for the overall estimate.
Question 7 · Easy
In an experiment on plant growth, a researcher uses 24 plants from 6 different species (4 of each). She randomly assigns 2 plants of each species to receive fertilizer and 2 to a control group. What design is this?
- A
  Completely randomized design
  Why not A: A completely randomized design assigns all units to treatments without blocking; here species is used as a blocking factor.
- B
  Randomized block design with species as the blocking variableCorrect
- C
  Matched pairs design
  Why not C: Matched pairs uses pairs of 2 units; here each block has 4 units (2 per treatment).
- D
  Stratified random sample
  Why not D: Stratified sampling is for surveys, not experiments; experiments use blocking.
Explanation
This is a randomized block design. The researcher groups plants by species (block) to control for species-to-species variation, then randomly assigns plants within each block to treatments. This is analogous to stratified sampling but in an experimental context.

Key takeaway
Blocking in experiments is analogous to stratification in sampling: it reduces variability by grouping similar units together before random assignment.
Question 8 · Easy
A call center randomly selects 50 of its 500 daily calls to monitor for quality. This is an example of what type of sample, and what is its main advantage over a census?
- A
  Convenience sample; easier to obtain
  Why not A: A random selection of records is not a convenience sample; convenience samples use whoever is easily accessible.
- B
  Simple random sample; it reduces cost and time while providing unbiased estimatesCorrect
- C
  Cluster sample; entire groups of calls are selected
  Why not C: Cluster sampling would select entire shifts or batches of calls, not 50 individual calls randomly.
- D
  Systematic sample; it eliminates variability
  Why not D: No $k$ is given; every call has an equal chance of selection, making this an SRS. Sampling does not eliminate variability.
Explanation
Randomly selecting 50 calls from 500 gives every call an equal chance of selection — a simple random sample. Advantages over a full census: lower cost, faster, practical when monitoring all calls is impossible, and it still provides unbiased estimates of the population.

Key takeaway
A simple random sample gives every unit an equal chance of selection and is more practical than a full census for large populations.
Question 9 · Medium
A survey asks: "Do you agree that the city council wastes taxpayer money on unnecessary projects?" What problem does this question have?
- A
  The question does not use a random sampling method.
  Why not A: The problem is with the question wording, not the sampling method.
- B
  The question is too short to collect meaningful data.
  Why not B: Length is not the issue; leading language is.
- C
  The question is leading and will likely produce response bias toward agreement.Correct
- D
  The question is subject to nonresponse bias.
  Why not D: Nonresponse bias is about who refuses to answer; here the issue is the phrasing that pushes toward a specific answer.
Explanation
The phrase "wastes taxpayer money on unnecessary projects" is emotionally loaded and leads respondents toward agreeing. This is a leading question, a form of response bias. A neutral version would ask: "How would you evaluate the city council's spending?"

Key takeaway
Leading questions create response bias by nudging respondents toward a particular answer through loaded or biased wording.
Question 10 · Medium
An experiment randomly assigns participants to one of three diets. After 8 weeks, mean weight loss is compared across groups. Which principle of experimental design ensures that unmeasured variables (like initial metabolism) are evenly distributed among the groups?
- A
  Replication
  Why not A: Replication means using enough subjects to detect real effects; it does not specifically distribute confounds across groups.
- B
  Control
  Why not B: Control refers to holding other variables constant or using a comparison group, not to distributing confounders.
- C
  Random assignmentCorrect
- D
  Blinding
  Why not D: Blinding reduces evaluator or subject bias but does not distribute confounding variables.
Explanation
Random assignment distributes known and unknown confounding variables (like metabolism, motivation, baseline health) approximately equally across treatment groups by chance. This is why randomized experiments can establish causal conclusions, unlike observational studies.

Key takeaway
Random assignment balances confounding variables across groups, allowing causal inference from experimental results.
Question 11 · Hard
A researcher observes that students who eat breakfast score higher on standardized tests than those who skip breakfast, using data from a voluntary survey. She concludes that eating breakfast improves test scores. Why is this an invalid causal conclusion?
- A
  The sample size from a voluntary survey is always too small.
  Why not A: Sample size may or may not be an issue; the fundamental problem is study design.
- B
  The data are observational; students were not randomly assigned to eat or skip breakfast, so confounding variables may explain the difference.Correct
- C
  Test score differences between groups cannot be measured accurately.
  Why not C: Test scores are a standard, measurable outcome; measurement accuracy is not the issue.
- D
  Voluntary surveys always overrepresent students who skip breakfast.
  Why not D: There is no reason to believe the direction of voluntary response bias here; and this is not the main problem.
Explanation
This is an observational study with voluntary response. Students who eat breakfast may differ in other ways (family stability, sleep, socioeconomic status) from those who skip it. These confounders, not breakfast itself, could cause higher scores. Only a randomized experiment can establish causation.

Key takeaway
In observational studies, confounding variables prevent causal conclusions even when an association exists.
Question 12 · Hard
A researcher conducts an experiment on pain relief using a matched pairs design. Each of 25 participants tries both Drug A and Drug B on different days, with order randomly assigned. What is the primary advantage of this design over a completely randomized design?
- A
  It eliminates the placebo effect entirely.
  Why not A: Matched pairs does not eliminate the placebo effect; blinding would address this.
- B
  It reduces variability by controlling for individual differences in pain sensitivity.Correct
- C
  It doubles the effective sample size by using each person twice.
  Why not C: While each person provides two measurements, the analysis uses pairwise differences ( $n = 25$ ), not $n = 50$ .
- D
  It allows the researcher to test more than two treatments simultaneously.
  Why not D: Matched pairs specifically compares two conditions; testing more treatments uses other designs.
Explanation
In a matched pairs design, each participant serves as their own control. Individual differences in pain tolerance are eliminated from the comparison because each person experiences both treatments. The analysis focuses on within-person differences, greatly reducing variability and increasing the sensitivity of the test.

Key takeaway
Matched pairs controls for individual variability by having each subject experience both conditions; the analysis is on pairwise differences.