Introducing Statistics: Are My Results Unexpected
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AP Statistics › Introducing Statistics: Are My Results Unexpected
A hospital states that 5% of patients experience nausea after a certain medication. A doctor gives the medication to 60 randomly selected patients and 7 report nausea. The doctor expected about 3 but observed 7. Is the result unexpected, given random variation in 60 patients?
Yes; it is impossible to see 7 nausea cases out of 60 if the true rate is 5%.
No; since 60 is not at least 200, the outcome cannot be considered surprising.
No; because 7 is close to 3, it is not surprising.
Yes; 7 out of 60 is unusually high if the true nausea rate is 5%.
No; because nausea is random, any number of patients could report it without being unexpected.
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.
A website claims that 10% of visitors click a certain ad. During one hour, 200 independent visitors arrive; the site manager expects about 20 clicks but observes 35 clicks. Assuming the 10% rate is correct, is this result unexpected given typical sampling variability for a sample of 200?
Yes; with 200 visitors, the click rate usually stays close to 10%, so 35 clicks (17.5%) is unusual though possible.
Yes; it is impossible to get 35 clicks if the true rate is 10%.
No; the expected number of clicks is 20, so any result between 0 and 40 should be considered expected.
No; since the sample is large, any deviation from 10% is expected.
No; because 35 is only 15 above 20, it should happen frequently with 200 visitors.
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.
A cereal company claims that 10% of its boxes contain a coupon. A student buys 8 boxes and finds coupons in 5 of them. The student expected about 0 or 1 coupon, but observed 5. Is the result unexpected, given that coupon placement is random and each box has a 10% chance of containing a coupon?
Yes; 5 out of 8 is unusually high if the true chance is 10%, so it would be surprising.
Yes; it is impossible to get 5 coupons out of 8 if the true chance is 10%.
No; because the student did not buy at least 30 boxes, the result cannot be considered unusual.
No; with only 8 boxes, results can vary a lot, and 5 coupons is not necessarily surprising.
No; because 10% does not mean the student must get exactly 0.8 coupons, any result is equally likely.
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.
An online course historically has a 60% completion rate. In a new cohort of 15 independently enrolled students, the instructor expects about 9 completions but observes 3 completions. Assuming the 60% rate still applies, is this result unexpected given the variability in a group of 15?
No; because the sample size is small, the completion rate is equally likely to be any value from 0% to 100%.
No; 3 completions is close enough to 9 that it should be considered expected.
Yes; 3 completions is impossible if the true completion rate is 60%.
No; since 60% is greater than 50%, observing fewer than half completing is still expected.
Yes; 3 out of 15 (20%) is much lower than 60%, making it unusual though possible by chance.
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.
A coin is advertised as fair. A student flips it 30 times, expecting about 15 heads, but gets 22 heads. Assuming the coin is actually fair and flips are independent, is this result unexpected given natural variability in 30 flips?
No; any result above 15 heads should be expected because randomness tends to produce more heads than tails.
Yes; 22 heads is impossible if the coin is fair, so the coin must be biased.
Yes; 22 heads in 30 flips is somewhat unusual for a fair coin, though it can occur by chance.
No; because the coin is fair, all head counts from 0 to 30 are equally likely.
No; 22 heads is only 7 away from 15, so it is a typical outcome for 30 flips.
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.
A factory’s historical defect rate is 2%. A quality inspector randomly checks 50 items from today’s production, expecting about 1 defect, but finds 4 defects. Assuming the true defect rate is still 2% and items are independent, is the result unexpected due to random chance?
Yes; it is impossible to observe 4 defects if the true defect rate is 2%.
Yes; 4 out of 50 (8%) is several times the expected rate and is unusual, though not impossible.
No; because the expected count is 1, any count from 0 to 5 is equally likely.
No; small samples always produce extreme percentages, so 8% should be expected.
No; 4 defects is close to 1 defect, so it is a typical outcome.
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.
A genetics lab expects that 50% of seeds from a certain cross will show a purple trait. A student grows 12 seeds, expecting about 6 purple, but observes 11 purple. Assuming the 50% model is correct and each seed is independent, is this result unexpected given sample-to-sample variability?
No; because the expected value is 6, any outcome within 6 of 6 should be considered expected.
No; with only 12 seeds, extreme results like 11 purple should be expected most of the time.
No; since 50% means outcomes are balanced, 11 purple is just as likely as 6 purple.
Yes; 11 purple is impossible if the true probability is 50%.
Yes; getting 11 out of 12 when the chance is 50% is quite unusual, though it can happen by chance.
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.
A city claims that 30% of residents use public transit at least once per week. A poll of 20 randomly selected residents expects about 6 transit users, but 0 of the 20 report using transit weekly. Assuming the 30% claim is true and responses are independent, is this result unexpected due to random variability?
No; because the sample size is only 20, the observed proportion can be anything without being unexpected.
No; since the expected number is 6, any result from 0 to 12 is typical.
Yes; it is impossible to observe 0 if the true rate is 30%.
No; in any poll, getting 0 is expected because someone might not use transit.
Yes; 0 out of 20 is very low if the true rate is 30%, so it is unusual though possible.
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.
A website A/B test assumes that if two page designs are equally effective, each visitor is equally likely to click either design’s button, so the probability a visitor clicks Design A is 0.50. In a random sample of 40 visitors shown both designs, 30 click Design A. The team expected about 20 clicks for A but observed 30. Is the result unexpected, given random variation with 40 visitors?
No; 30 out of 40 is close enough to 20 out of 40 that it is not surprising.
Yes; 30 out of 40 is far from 20 out of 40 and would be surprising if $p=0.50$.
No; since 40 is not a large sample, the outcome cannot be considered unusual.
Yes; it is impossible to get 30 clicks for A if $p=0.50$.
No; because $p=0.50$ means Design A will win half the time, and 30 is half of 40.
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.
A delivery service claims that 90% of its packages arrive on time. A customer tracks 15 packages and finds that 11 arrive on time. The customer expected about 14 on-time deliveries but observed 11. Is the result unexpected, given the variability with only 15 packages?
Yes; 11 out of 15 is low enough to be surprising if the true on-time rate is 90%.
No; because the sample size is under 30, the result cannot be considered unusual.
No; 11 out of 15 is close enough to 90% that it is not surprising.
No; because 90% is not 100%, any result is equally likely in 15 packages.
Yes; it is impossible for only 11 packages to be on time if the true rate is 90%.
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.