AP Statistics Quiz: Introducing Statistics Are My Results Unexpected

What this quiz covers

This quiz focuses on Introducing Statistics Are My Results Unexpected, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Yes; 7 out of 60 is unusually high if the true nausea rate is 5%. (correct answer)
2. No; because 7 is close to 3, it is not surprising.
3. No; because nausea is random, any number of patients could report it without being unexpected.
4. Yes; it is impossible to see 7 nausea cases out of 60 if the true rate is 5%.
5. No; since 60 is not at least 200, the outcome cannot be considered surprising.

Explanation: The skill is determining unexpectedness in binomial settings, AP Statistics intro. n=60 patients, p=0.05, expected 3, SD ~1.69, typical 0-6. 7 is ~2.4 SD above, probability ~3%, unexpected. Distractor B says 7 close to 3, but relative to variability it's not. Mini-lesson: Check SD distance; >2 SD suggests surprise, rare by chance.

Question 2

1. No; because 35 is only 15 above 20, it should happen frequently with 200 visitors.
2. Yes; with 200 visitors, the click rate usually stays close to 10%, so 35 clicks (17.5%) is unusual though possible. (correct answer)
3. No; since the sample is large, any deviation from 10% is expected.
4. Yes; it is impossible to get 35 clicks if the true rate is 10%.
5. No; the expected number of clicks is 20, so any result between 0 and 40 should be considered expected.

Explanation: This question tests understanding of sampling variability with a larger sample size. With 200 visitors and a 10% click rate, we expect about 20 clicks, but 35 were observed (17.5% click rate). With larger samples, the observed proportion tends to stay closer to the true proportion compared to smaller samples. The standard deviation for this scenario is about 4.2, so 35 clicks is roughly 3.5 standard deviations above the mean—a very unusual result. While not impossible, this outcome would occur less than 0.1% of the time if the true rate is 10%. The key lesson is that larger samples reduce variability, making deviations from the expected value more meaningful and surprising.

Question 3

1. No; with only 8 boxes, results can vary a lot, and 5 coupons is not necessarily surprising.
2. Yes; 5 out of 8 is unusually high if the true chance is 10%, so it would be surprising. (correct answer)
3. No; because 10% does not mean the student must get exactly 0.8 coupons, any result is equally likely.
4. Yes; it is impossible to get 5 coupons out of 8 if the true chance is 10%.
5. No; because the student did not buy at least 30 boxes, the result cannot be considered unusual.

Explanation: This question assesses the skill of determining if observed results are unexpected in a binomial setting, part of introducing statistics in AP Statistics. Here, with n=8 trials and p=0.10 probability of success, the expected number of coupons is 0.8, and the standard deviation is about 0.85, so typical results range from roughly 0 to 2 or 3. Observing 5 coupons is about 5 standard deviations above the mean, which is highly unusual and thus unexpected. A common distractor, like choice A, suggests that small samples always vary too much for any result to be surprising, but this ignores that even small samples have limited variability, and extreme outcomes like this have very low probability. In a mini-lesson on judging surprise: calculate the mean np and standard deviation sqrt(np(1-p)), then see if the observed value is more than 2-3 standard deviations away; if so, it's surprising as it would rarely occur by chance alone.

Question 4

1. No; 3 completions is close enough to 9 that it should be considered expected.
2. Yes; 3 out of 15 (20%) is much lower than 60%, making it unusual though possible by chance. (correct answer)
3. No; because the sample size is small, the completion rate is equally likely to be any value from 0% to 100%.
4. Yes; 3 completions is impossible if the true completion rate is 60%.
5. No; since 60% is greater than 50%, observing fewer than half completing is still expected.

Explanation: This question tests understanding of how dramatically an outcome can deviate from expectation in small samples. With a 60% completion rate and 15 students, we expect about 9 completions, but only 3 were observed (20% rate). The probability of getting 3 or fewer completions out of 15 when the true rate is 0.6 is extremely small (less than 0.2%), making this a highly unexpected result. Even accounting for the smaller sample size, this outcome is far enough from expectation to suggest something unusual—perhaps the new cohort differs from historical students or the course has changed. Students should understand that while small samples show more variability, there are still limits to what random chance alone typically produces.

Question 5

1. No; 22 heads is only 7 away from 15, so it is a typical outcome for 30 flips.
2. Yes; 22 heads in 30 flips is somewhat unusual for a fair coin, though it can occur by chance. (correct answer)
3. No; because the coin is fair, all head counts from 0 to 30 are equally likely.
4. Yes; 22 heads is impossible if the coin is fair, so the coin must be biased.
5. No; any result above 15 heads should be expected because randomness tends to produce more heads than tails.

Explanation: This question tests understanding of coin flip variability with a moderate sample size. With a fair coin flipped 30 times, we expect about 15 heads, but 22 were observed. The probability of getting 22 or more heads in 30 flips of a fair coin is approximately 1.6%, which is small enough to be considered unusual. While this could happen by chance with a fair coin, it's rare enough to raise questions about whether the coin might be biased toward heads. Students should understand that "unexpected" results can still occur with truly fair processes, but their rarity makes us consider alternative explanations. The key is distinguishing between impossible outcomes and unlikely but possible ones.

Question 6

1. No; 4 defects is close to 1 defect, so it is a typical outcome.
2. Yes; 4 out of 50 (8%) is several times the expected rate and is unusual, though not impossible. (correct answer)
3. No; because the expected count is 1, any count from 0 to 5 is equally likely.
4. Yes; it is impossible to observe 4 defects if the true defect rate is 2%.
5. No; small samples always produce extreme percentages, so 8% should be expected.

Explanation: This question examines whether finding 4 defects in 50 items is unexpected when the historical rate is 2%. With a 2% defect rate, we'd expect about 1 defect in 50 items, so observing 4 is notably higher. The probability of finding 4 or more defects when the true rate is 0.02 is approximately 2.4%, which is small enough to be considered unusual. This result suggests either an unlucky sample or a potential quality control issue. Students should recognize that while 4 defects could occur by chance alone, it's rare enough to warrant attention. The distinction between "impossible" and "unlikely but possible" is crucial for proper statistical thinking.

Question 7

1. Yes; getting 11 out of 12 when the chance is 50% is quite unusual, though it can happen by chance. (correct answer)
2. No; since 50% means outcomes are balanced, 11 purple is just as likely as 6 purple.
3. No; with only 12 seeds, extreme results like 11 purple should be expected most of the time.
4. Yes; 11 purple is impossible if the true probability is 50%.
5. No; because the expected value is 6, any outcome within 6 of 6 should be considered expected.

Explanation: This question tests understanding of variability in small samples with a 50% probability. When growing 12 seeds with a 50% chance of showing purple, we expect about 6 purple seeds, but 11 were observed. The probability of getting 11 or more purple seeds out of 12 when the true probability is 0.5 is very small (about 0.3%), making this an extremely unusual result. Even with small samples where we expect more variability, this outcome is far enough from expectation to be considered unexpected. Students should understand that while small samples do show more variability than large ones, there are still limits to what we'd reasonably expect to see by chance. This result would prompt questions about whether the genetic model is correct.

Question 8

1. No; in any poll, getting 0 is expected because someone might not use transit.
2. Yes; 0 out of 20 is very low if the true rate is 30%, so it is unusual though possible. (correct answer)
3. No; since the expected number is 6, any result from 0 to 12 is typical.
4. Yes; it is impossible to observe 0 if the true rate is 30%.
5. No; because the sample size is only 20, the observed proportion can be anything without being unexpected.

Explanation: This question examines an extreme outcome where 0 out of 20 residents report using transit when 30% is expected. With a 30% usage rate, we'd expect about 6 out of 20 to use transit, so observing 0 is a complete absence where we expected a substantial number. The probability of getting 0 successes in 20 trials when the probability is 0.3 is extremely small (about 0.0001%), making this result highly unexpected. This outcome is so unlikely that it strongly suggests either the 30% claim is incorrect, the sample wasn't truly random, or something else unusual occurred. Students should recognize that while technically possible, some outcomes are so improbable that they warrant serious questioning of our assumptions.

Question 9

1. No; 30 out of 40 is close enough to 20 out of 40 that it is not surprising.
2. Yes; 30 out of 40 is far from 20 out of 40 and would be surprising if $p=0.50$. (correct answer)
3. No; because $p=0.50$ means Design A will win half the time, and 30 is half of 40.
4. Yes; it is impossible to get 30 clicks for A if $p=0.50$.
5. No; since 40 is not a large sample, the outcome cannot be considered unusual.

Explanation: The skill involves deciding if results are unexpected given binomial variability, part of AP Statistics' introductory statistics. With n=40 visitors, p=0.50, expected 20 clicks, SD ~3.16, typical 14-26. 30 is >3 SD above, probability <0.1%, unexpected. Distractor A claims 30 is close enough, but it's far outside typical range. Mini-lesson: Calculate mean and SD; if observed > mean + 3 SD, it's surprising as chance alone rarely produces it.

Question 10

1. No; 11 out of 15 is close enough to 90% that it is not surprising.
2. Yes; 11 out of 15 is low enough to be surprising if the true on-time rate is 90%. (correct answer)
3. No; because 90% is not 100%, any result is equally likely in 15 packages.
4. Yes; it is impossible for only 11 packages to be on time if the true rate is 90%.
5. No; because the sample size is under 30, the result cannot be considered unusual.

Explanation: This question tests assessing unexpected binomial variability, from AP Statistics. n=15 packages, p=0.90, expected 13.5, SD ~1.16, typical 11-15. 11 is ~2.2 SD below, probability ~4%, unexpected. Distractor A claims close enough, but it's outside typical. Mini-lesson: If < mean - 2 SD, surprising as low probability.

Question 11

1. Yes; 41 clicks is impossible if the true proportion is 0.50.
2. No; with $n=100$, getting 41 instead of 50 can happen due to random variation. (correct answer)
3. Yes; because 41 is not equal to the expected 50, it is unexpected.
4. No; because 50% means the number of clicks must be exactly 50 out of 100.
5. Yes; because the expected value is 50, any value below 50 is unexpected.

Explanation: This question tests understanding of sampling variability for proportions with moderate sample sizes. With a true proportion of 50% and n=100, we expect about 50 clicks, but the actual count will vary due to sampling. Getting 41 clicks represents a sample proportion of 41%, which is 9 percentage points below the claimed 50%. While this deviation might seem notable, it's well within the range of normal sampling variability for n=100. The standard deviation for this scenario is about 5, so counts between roughly 40-60 are quite plausible. The distractors incorrectly suggest that any deviation from 50 is impossible or unexpected, misunderstanding the nature of random sampling.

Question 12

1. Yes—because the mean decreased, the review game must be ineffective; chance variation cannot cause a decrease.
2. No—because with only 8 students and one unusually low score, the sample mean can fall below 75% even if the game helps. (correct answer)
3. Yes—because a single low score cannot noticeably change a mean.
4. No—because random selection ensures the mean is always close to 80% when the game is used.
5. Yes—because any sample mean below 75% is essentially impossible if the true mean is 80%.

Explanation: This question tests recognition that outliers can substantially affect small sample means. With only 8 students, one unusually low score (40% from an absent student) can pull the mean down from the expected 80% to 73%, even if the review game typically helps. This result is not unexpected because small samples are vulnerable to extreme values. The incorrect answers either claim any decrease proves ineffectiveness, misunderstand how outliers affect means, or incorrectly assume random selection eliminates variability. When interpreting means from small samples, always consider whether unusual observations might be driving the results rather than representing a true effect.

Question 13

1. Yes; because 60% is greater than 30%, it is impossible if the true win probability is 30%.
2. No; with 15 plays, results can vary, and 9 wins could occur by chance even if the true win probability is 30%, though it may be unusual. (correct answer)
3. Yes; because random chance should produce exactly 30% wins every time.
4. No; because every number of wins from 0 to 15 is equally likely.
5. Yes; because the result is unusual, it proves the booth's stated probability must be 60%.

Explanation: This question assesses understanding of binomial variability with small trials. Winning 9 times out of 15 plays (60%) when the stated probability is 30% seems like a large difference, but with only 15 trials, such variation can occur by chance. While this outcome is somewhat unusual, it's not impossible - the binomial distribution shows that getting 9 or more successes out of 15 when p=0.3 has a calculable, non-zero probability. The incorrect options either claim this is impossible or misunderstand probability concepts. When judging whether results are unexpected, we must consider that with limited trials, observed proportions can deviate substantially from true probabilities due to chance alone, though extreme results may warrant further investigation.

Question 14

1. No—because with only 20 users, the observed proportion can easily be below 40% due to random variation. (correct answer)
2. Yes—because a helpful feature cannot produce a sample proportion below the old 40% rate.
3. Yes—because 30% is unusual, so it proves the feature reduces workout completion.
4. No—because random assignment guarantees the sample proportion will be at least 45% when the feature works.
5. Yes—because the sample size is 20, which is large enough that chance variation is negligible.

Explanation: This question assesses whether a sample proportion below expectations is surprising given sampling variability. With only 20 users, the sample proportion of 30% (6 out of 20) completing workouts is not unexpected even if the true rate is 45%. Small samples naturally have high variability in proportions - getting 6 successes instead of the expected 9 can easily occur by chance. The incorrect answers either claim this result is impossible if the feature works, misunderstand random assignment, or incorrectly assume 20 is large enough to eliminate sampling variability. When evaluating proportions from small samples, remember that observed proportions can vary considerably from the true population proportion due to random chance.

Question 15

1. No; 68% is expected because the sample size is small, so the sample proportion can vary noticeably. (correct answer)
2. Yes; 68% is unexpected because any deviation from 80% indicates the true rate is not 80%.
3. Yes; 68% is impossible if the true rate is 80%.
4. No; 68% is expected because 80% means exactly 20 out of 25 should finish.
5. No; 68% is not surprising because random samples always produce results close to the true rate.

Explanation: AP Statistics' introducing statistics skill here involves assessing if a result is unexpected due to chance in sampling. With 80% true rate and n=25, SD=0.08, expected proportions range from 64% to 96% (2 SDs). The 68% is only 1.5 SDs away, so not surprising. Distractor B claims any deviation means the rate isn't 80%, ignoring normal variability. Mini-lesson: Compute SD and evaluate if observed is within 2-3 SDs of p; if yes, it's expected chance variation. This teaches that samples fluctuate, and small n amplifies that without implying a different true rate.

Question 16

1. Yes; 8 heads is unexpected because it is impossible with a fair coin.
2. No; 8 heads is not surprising because with only 10 flips, getting 8 heads can happen by chance. (correct answer)
3. Yes; 8 heads is unexpected because any result other than exactly 5 heads means the coin is unfair.
4. No; 8 heads is expected because a fair coin favors heads half the time.
5. No; 8 heads is not surprising because all outcomes from 0 to 10 heads are equally likely.

Explanation: This AP Statistics question on introducing statistics evaluates if coin flip results are unexpected given fair probability and small n. For p=0.5, n=10, SD≈0.158, typical heads proportions 18% to 82% (2 SDs). 80% is 1.9 SDs away, not surprising for small flips. Distractor E says all outcomes equally likely, but binomial probabilities vary, with extremes less likely. Mini-lesson: Use SD to gauge surprise; under 2 SDs often means expected, especially in small samples where variability is high. This highlights that deviations don't always indicate unfairness.

Question 17

1. Yes; 46.7% is unexpected because it is far from 65% for a sample of 30. (correct answer)
2. No; 46.7% is expected because sampling at a transit hub guarantees a lower transit preference.
3. No; 46.7% is not surprising because small samples can show substantial variation from 65%.
4. Yes; 46.7% is impossible if the true rate is 65%.
5. No; 46.7% is not surprising because percentages are always imprecise.

Explanation: This question in AP Statistics introduces statistics by asking if a sample is unexpected given variability around a true proportion. For 65% preference with n=30, SD ≈0.087, typical range is about 48% to 82% (2 SDs). The 46.7% is roughly 2.1 SDs below, just enough to be considered unexpected for this context. Distractor C says small samples show substantial variation, but here the deviation is still notable relative to SD. Mini-lesson: Judge surprise by finding SD = sqrt(p(1-p)/n) and checking SD distance; borderline cases like 2+ SDs often indicate surprise, especially if far in context. This builds intuition for when variability can't explain results.

Question 18

1. Yes; a sample mean of 6.5 minutes cannot occur by chance if the true mean is 4 minutes.
2. No; with only 8 customers, the sample mean can vary noticeably from day to day, so 6.5 minutes could plausibly occur by chance. (correct answer)
3. Yes; because the sample mean is above 4, it must indicate the true mean is above 4.
4. No; because the manager sampled randomly, the sample mean must equal the population mean exactly.
5. Yes; because the sample size is 8, the sample mean should be extremely stable and close to 4 minutes.

Explanation: This question assesses understanding of how sample means vary, especially with small samples. With only 8 customers, the sample mean can fluctuate considerably from day to day due to random variation. An observed mean of 6.5 minutes when expecting 4 minutes represents a 2.5-minute difference, which is plausible given the small sample size. The distractors incorrectly suggest either that this variation is impossible or that sample means should exactly match population means. When evaluating whether a sample mean is unexpected, we must recognize that smaller samples produce more variable estimates, making larger deviations from the population mean more likely to occur by chance alone.

Question 19

1. No; with a sample of 20, sample proportions can vary a lot, so a result like 40% could happen by chance even if the true proportion is about 70%. (correct answer)
2. Yes; because 40% is different from 70%, it is essentially impossible to occur by chance if the true proportion is 70%.
3. No; any sample proportion is equally likely, so 40% is just as expected as 70%.
4. Yes; because the sample size is 20, the sample proportion should be very close to 70% almost every time.
5. Yes; because the result is unusual, it proves the true proportion cannot be 70%.

Explanation: This question tests understanding of sampling variability when estimating proportions from small samples. With only 20 students, we expect considerable variation in sample proportions even if the true population proportion is 70%. The observed 40% (8 out of 20) represents a difference of 30 percentage points from expectation, but such deviations are not unusual with small samples. The incorrect options either claim this result is impossible (which ignores sampling variability) or misunderstand how random sampling works. When judging whether a result is surprising, we must consider both the size of the difference and the sample size - smaller samples naturally produce more variable results.

Question 20

1. Yes; because 83% is below 95%, it is essentially impossible if the true on-time rate is 95%.
2. No; with 30 deliveries, some variation is expected, and 25 on-time could occur by chance even if the true rate is 95%, though it may raise questions. (correct answer)
3. Yes; because a random sample should match the company's claim almost exactly.
4. No; because the true on-time rate does not affect the sample results in any way.
5. Yes; because the result is unusual, it proves the company is lying about the 95% rate.

Explanation: This question assesses understanding of sampling variability for proportions with moderate sample sizes. Observing 25 out of 30 deliveries on time (83%) when expecting 95% represents a notable deviation, but with a sample of 30, such variation can occur by chance. While this result might raise questions about the company's claim, it's not impossible under random sampling. The incorrect options either claim this is impossible or suggest that unusual results definitively prove claims false. When evaluating whether results are unexpected, we must balance the size of the deviation against the sample size - even with 30 observations, random variation can produce results that differ meaningfully from the true proportion, though such deviations become less likely as sample size increases.