AP Statistics Discussion Questions, by Topic

Why AP Statistics needs argument, not just arithmetic

Most AP Statistics practice trains one skill at a time: compute a standard deviation, carry out a test, interpret an interval. The exam's free response questions ask for something else. They hand you a scenario and ask you to justify a conclusion, which means weighing evidence and defending a judgment call, not retrieving a formula.

Discussion is the format that trains judgment directly, because a good discussion question has a correct underlying principle and a genuinely contestable application of it. A worksheet question has one right number. The forty-two questions below do not, or at least not one that every reasonable, well-informed student will reach the same way.

Use them as an opener before a unit, as a full period of review before a test, or as five-minute breaks inside a lecture when attention drops. For scenario-based practice with a single worked solution instead of a debate, see scope of inference practice and experimental design practice.

Study design and causation

Every AP Statistics student can recite the rule: random assignment supports a causal claim, random selection supports generalizing it to a population. The argument starts one level up, when a real study only gets one of the two, gets both imperfectly, or gets both cleanly but a decision cannot wait for a better one. Experiments vs observational studies and can you generalize these results cover the rule itself; these questions live in the gap the rule does not resolve for you.

1. A large observational study finds that people who take a daily supplement live measurably longer. Should a doctor recommend the supplement on this evidence alone, or wait for a randomized trial? Name a gap in life expectancy that would be large enough for you to act on the observational finding anyway.
2. A school randomly assigns half its incoming class to a study-skills seminar and the treated half's GPA rises more than the other half's. A parent whose child was assigned to the control group demands the seminar for their child too, since it "obviously works." Is refusing that request defensible?
3. A drug cannot ethically be randomly assigned to teenagers to test whether it stunts growth, so all the evidence on the question is observational. Does the impossibility of ever running the clean experiment mean the observational conclusion stays provisional forever, or can enough converging observational studies substitute for the experiment nobody is allowed to run?
4. In a randomized experiment, 12 of 60 people assigned to the treatment group drop out before it ends, and nobody drops out of the control group. Does the uneven dropout undo the causal conclusion, weaken it, or leave it intact, and where exactly is the line?
5. A completely randomized, double-blind trial of a new sleep aid is run entirely on retired volunteers recruited from one clinic. Is the causal claim here strong or weak? Is the generalization strong or weak? Can a single study genuinely be excellent on one axis and close to worthless on the other?
6. Is it possible for a well-designed experiment to be less useful to a policymaker than a well-designed observational study? Argue for a specific case where you would act on the correlational finding instead of waiting for a randomized one.
7. Two teams study the same claim: one runs a randomized experiment on a small pool of volunteers, the other runs an observational study on records covering an entire population. Which result would you trust to set a policy meant for everyone, and why doesn't studying literally everyone settle the causal question by itself?

Sampling and bias

Bias is a property of a method, not of a person's intentions, which is why a well-meaning survey can be badly biased and a badly motivated one can, by luck, land close to the truth. The most common wrong turn in this unit is treating sample size as a stand-in for sample quality; does a bigger sample fix bias and how to identify the type of bias lay out why it is not, and these questions test whether that separation survives contact with a messier example.

1. A city mails a satisfaction survey to a random sample of 5,000 households and gets 200 responses back. Is this a good estimate of citywide satisfaction, a biased one, or is there not yet enough information to say?
2. A news app collects a genuinely random sample of its own users, all of whom opted in to political-news alerts, and reports the results as a poll of the public. What population, if any, does this result actually describe?
3. Would you rather bet real money on a poorly randomized sample of 50,000 people or a carefully randomized sample of 500?
4. A convenience sample of patients at one hospital happens, by coincidence, to match the county's age and income distribution exactly. Is the resulting estimate unbiased, or is that the wrong question to ask about a single sample?
5. Every voluntary response sample this class has read about turned out to be biased toward the group with the strongest feelings. Can you design a voluntary response sample that avoids that particular bias, even in principle, or is it built into what "voluntary response" means?
6. Splitting a sample by an irrelevant trait, like the first letter of a last name, and splitting it by a relevant one, like grade level, both technically produce a stratified random sample. Should the term require that the strata matter, or is that a separate claim from the sampling mechanics?
7. Nonresponse is often assumed to bias a survey in one predictable direction, since people with strong feelings respond more. Can the direction of nonresponse bias be predicted before collecting the data, or only diagnosed after the fact? See selection bias and census vs sample survey for two of the mechanisms behind these questions.

Probability intuition

Correct probability answers routinely clash with strong gut instincts, and the disagreement in class is usually about which instinct deserves to be trusted, not about the arithmetic once it is laid out. Disjoint vs independent events and probability vs odds sort out the vocabulary; these questions test whether the vocabulary survives an intuition pump.

1. Two independent events each have probability 0.5. Is knowing the outcome of the first one ever useful information about the second, even in a loose, non-mathematical sense, given that the formal answer is no?
2. Two events are described as mutually exclusive. A student argues they must also be independent, since neither one "affects" the other. Is that a defensible use of the word independent, and exactly where does it break down?
3. A forecast gives a 70% chance of rain, and it does not rain. Was the forecast wrong?
4. In a class of 30 students, which is more surprising: that two people share a birthday, or that nobody does?
5. A gambler who just watched a roulette wheel land on red six times in a row bets heavier on black next. Is there any version of that reasoning that holds up, even if the standard gambler's fallacy framing does not?
6. A medical test is 99% accurate and comes back positive for a disease that only 1 in 10,000 people have. Is the patient more likely to have the disease or not, and why does the 99% figure feel like it should settle the question when it does not? Try probability of at least one as a warm-up before this one.
7. Two events are independent in the statistical sense. Does that mean they are unrelated in every sense an ordinary English speaker would mean by "related"?

What a p-value does and does not say

The p-value is one of the most consistently misstated numbers in an introductory statistics course, and students who can compute one correctly still often reach for the wrong sentence the moment they are asked to explain it out loud. What does a p-value mean and p-value vs alpha give the correct wording; use these questions to see whether the class can hold onto it under pressure. A p-value is never the probability that the null hypothesis is true, and none of the correct answers below should be read that way.

1. A p-value of 0.03 comes back for a new teaching method. A student says, "there's a 3% chance the method doesn't actually work." What exactly is wrong with that sentence, and can the class fix it in one sentence without losing anyone?
2. Is a p-value of 0.049 meaningfully different from a p-value of 0.051, given that the standard cutoff treats them as opposite conclusions?
3. Two studies test the same claim. One gets a p-value of 0.001 from a sample of 40; the other gets a p-value of 0.04 from a sample of 4,000. Which study gives stronger evidence of a real, useful effect?
4. If a result is not statistically significant, has the study proven the null hypothesis true?
5. A researcher runs 20 independent tests on the same dataset, and exactly one comes back significant at the 0.05 level. Should that one result be reported as a discovery?
6. Does a smaller p-value always mean a bigger, more important effect?
7. A significance level of 0.05 means roughly a 5% chance of a false alarm when the null hypothesis is actually true. Should every field use the same 0.05 cutoff, or should the number depend on what a false alarm costs? Bring in type 1 vs type 2 errors and how to choose a significance level before deciding.

Confidence intervals

A confidence interval reads like a statement about the one interval sitting in front of you, but it is actually a statement about the method that produced it, and this class has already met people who blur that line on purpose to sound more certain than the data supports. What 95% confidence means and confidence level vs confidence interval hold the precise wording.

1. "There is a 95% probability that the true mean falls inside this specific interval." Once the interval has actually been calculated from the one sample in hand, is that sentence true?
2. A pollster reports a candidate at 52% with a 3-point margin of error and calls it a clear lead. Is that a fair description of an interval running from 49% to 55%?
3. Would a 99% confidence interval or a 90% confidence interval be more useful to a doctor deciding whether a new treatment's effect is real? Does higher confidence always mean more useful?
4. Two different random samples from the same population produce two 95% confidence intervals that do not overlap at all. What, if anything, went wrong?
5. If 100 different random samples from the same population each produced their own 95% confidence interval, roughly how many of the 100 intervals would actually contain the true parameter, and is that number a guarantee for any single one of the 100?
6. A confidence interval for a difference between two groups contains zero. Does that mean the two groups are the same? Try two proportion confidence interval contains zero after the class settles this one.
7. Making a confidence interval narrower always sounds like an improvement. Describe a way of narrowing one that would actually make the result worse, not better.

Data ethics

The ethical weight of a study is not obvious from its statistics alone, and two designs that are equally rigorous can differ sharply in whether either one should have been run at all, or run the way it was. Correlation vs causation and scope of inference supply the technical vocabulary that these questions push on from the ethical side.

1. A randomized experiment could answer a valuable medical question, but only if a control group knowingly receives a treatment researchers already suspect does not work. Is a strong experimental design ever ethically identical to a pretext for withholding care?
2. A company changes its app for millions of users without telling any of them, measures the effect, and reports the result as an experiment. Does the lack of consent change whether the statistics are valid, whether they are usable, or neither?
3. A public health study wants a genuine random sample of an undocumented population to estimate need for services, which means finding and contacting people who have real reasons to avoid being found. Is refusing to attempt random sampling here the more ethical choice, even though it guarantees a weaker, less generalizable study?
4. A dataset scraped from social media can answer a real research question, but nobody in it agreed to be studied. Does the usefulness of the answer change your view of whether it should have been collected?
5. Is it more ethical to run an underpowered study that respects every constraint researchers face, or a well-powered one that cuts a corner on consent to get there?
6. A company anonymizes its user data before selling it for research, but a separate public dataset makes part of it possible to re-identify. Whose responsibility is that: the company that anonymized it, the buyer, or the publisher of the second dataset?
7. Misreading a p-value or a poll can cause real harm outside the classroom. When a statistic gets reported wrong, does the responsibility sit with the person who reported it, the person who misread it and acted on it, or both equally?

How to run the discussion

The format matters more than the question. A teacher who reads question 22 aloud and immediately calls on a volunteer gets one confident answer and silence from everyone who disagreed with it. A few habits change that.

Have students write a private answer before anyone speaks. Sixty seconds of silent writing forces a commitment before social pressure has a chance to flatten the room toward whoever spoke first. Then cold-call, and cold-call the quiet half of the room on purpose, since the students who write a different answer than the first speaker are usually the ones worth hearing from.

Withhold the correct framing until two genuinely different answers are on the table. A discussion that gets fixed after the first response was never a discussion. If the class reaches quick, confident agreement, that is the moment to introduce the counterexample you had ready, not to move on.

Require a restatement before a rebuttal. A student may not respond to another student's answer without first restating it in their own words, checked by the original speaker. This single rule kills most of the talking-past-each-other that makes a stats discussion feel unproductive.

Timebox each question to somewhere between four and eight minutes, including the ones with a clean correct answer, like most of the p-value group. Six of these in a fifty-minute period, with a short break for a written answer between them, uses the period well without turning into a lecture with occasional pauses.

Do not grade for the right answer. Grading the content of an opinion on data ethics, or on how large an observational gap has to be before you act on it, guarantees that students report the position that seems safest rather than the one they hold. If a grade is needed, grade for whether a student restated an opposing view accurately, which is checkable and does not reward picking the popular side.

Close each question with the sentence the class would defend in writing, not a verdict. For the questions with a real correct principle, such as the p-value and confidence interval groups, that sentence should match the standard wording; students can then rehearse writing it as a full justification on experimental design practice, p-value interpretation practice, or confidence interval practice. For the ethics questions, the closing sentence is often "reasonable people land in different places, for these specific reasons," and that is a complete, honest answer.