Introducing Statistics: Are My Results Unexpected - AP Statistics

1. A wildlife biologist expects that a certain lake's fish population is about 50% female. The biologist catches a random sample of 12 fish and finds 10 are female. The expected behavior is that the sample proportion should be near 50%, but the observed outcome is about 83% female. Is this result unexpected, given the variability that can occur with random samples of size 12?
   Answer: No; with a sample size of 12, getting 10 females could happen by chance even if the true proportion is 50%, though it may be somewhat unusual. This question examines understanding of binomial variability in small samples. Finding 10 females out of 12 fish (83%) when expecting 50% seems extreme, but with such a small sample, this outcome can occur by chance. The probability of getting 10 or more females out of 12 when p=0.5 is calculable and, while somewhat unusual, is not impossible. The incorrect answers either claim this is impossible under random sampling or misunderstand probability concepts like equally likely outcomes. When assessing whether a result is unexpected, we must consider that small samples can produce proportions quite far from the population value simply due to chance, though extreme results may still raise questions about our assumptions.
2. A biology lab manual states that 25% of a certain plant's seeds germinate under standard conditions. A student assumes the true germination rate is 25% and tests 80 seeds; 30 germinate (37.5%). The student wants to know whether this outcome is surprising due to random sampling variability with $n=80$ if the true rate is 25%. Is the result unexpected?
   Answer: Yes; 37.5% is unexpected because it is noticeably higher than 25% for a sample of 80. This AP Statistics question introduces assessing if results are unexpected from sampling variability. With 25% rate, n=80, SD≈0.048, typical 15.4% to 34.6% (2 SDs). 37.5% is 2.6 SDs above, unexpected. Distractor E says larger samples have more variability, but actually less relative variability. Mini-lesson: Use SD to measure distance; over 2.5 SDs indicates surprise, suggesting possible higher true rate. This method helps interpret experimental outcomes reliably.
3. A streaming service reports that 80% of users finish a particular episode once they start it. A researcher assumes the true completion rate is 80% and randomly samples 25 users who started the episode; 17 finished (68%). The researcher asks whether 68% is surprising due to chance variation with $n=25$ if the true rate is 80%. Is the result unexpected?
   Answer: No; 68% is expected because the sample size is small, so the sample proportion can vary noticeably. 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.
4. 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?
   Answer: Yes; 7 out of 60 is unusually high if the true nausea rate is 5%. 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.
5. A basketball player makes 70% of free throws in the long run. During one practice, she shoots 20 free throws and makes 10. She expected about 14 makes but observed 10. Is the result unexpected, given natural variability in 20 shots?
   Answer: Yes; making only 10 out of 20 would be surprisingly low if her true rate is 70%. This problem tests the ability to evaluate if results are unexpected given binomial variability, a key concept in AP Statistics' introduction to statistics. For n=20 free throws with p=0.70, the expected makes are 14, with a standard deviation of about 2.05, meaning typical outcomes fall between roughly 10 and 18. Getting only 10 makes is nearly 2 standard deviations below the mean, which is unusually low and unexpected. Choice C, a distractor, claims that variation in 20 shots isn't unusual for this deviation, but calculations show the probability is around 5% or less, making it surprising. Mini-lesson: To judge surprise, compare the observed count to the mean np; if it's beyond about 2 standard deviations (sqrt(np(1-p))), it's unexpected, as such results occur in fewer than 5% of similar trials by chance.
6. 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?
   Answer: Yes; 22 heads in 30 flips is somewhat unusual for a fair coin, though it can occur by chance. 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.
7. A delivery company claims that 95% of its packages arrive on time. A customer randomly tracks 30 recent deliveries and finds that 25 arrived on time (about 83%). The customer expected the on-time rate in the sample to be near 95%, but it was lower. Is the result unexpected, considering sampling variability with a sample of 30 deliveries?
   Answer: 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. 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.
8. A factory's long-run defect rate for a certain part is about 2%. A quality engineer randomly inspects 500 parts from one day's production and finds 20 defective (4%). The engineer wants to know whether 4% is surprising for $n=500$ if the true defect rate is 2%, considering typical sampling variability. Is the result unexpected?
   Answer: Yes; 4% is unexpected because a sample of 500 should usually be quite close to 2%. Under AP Statistics' introducing statistics, this question checks understanding of whether results are unexpected based on sampling variability. With a 2% defect rate and n=500, SD is about 0.006, so expected proportions are typically 0.8% to 3.2% (2 SDs range). The 4% is over 3 SDs away, making it unexpected from variability alone. Choice A distracts by saying doubling is common in large n, but actually large n reduces variability, making doubles rare. Mini-lesson on judging surprise: Calculate SD and compare the deviation in SD units; over 2-3 SDs flags surprise, signaling possible shifts in the true rate. This method provides a quantitative way to assess sampling outcomes pedagogically.
9. 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?
   Answer: Yes; with 200 visitors, the click rate usually stays close to 10%, so 35 clicks (17.5%) is unusual though possible. 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.
10. A cereal company advertises that 40% of boxes contain a "bonus card." A teacher buys 10 boxes for a class activity, expecting about 4 bonus cards, but finds 9 bonus cards. Assuming the 40% rate is accurate and each box is independent, is this result unexpected given natural sample-to-sample variability?
    Answer: Yes; 9 out of 10 is very unusual if the true chance is 40%, though it is not impossible. This question tests understanding of whether an observed outcome is unexpected given natural sampling variability. With a 40% chance of bonus cards, we'd expect about 4 out of 10 boxes to contain them, but the teacher found 9. While this is much higher than expected, we need to consider whether such an outcome could reasonably occur by chance alone. Getting 9 or more successes out of 10 trials when the probability is 0.4 has a very low probability (less than 2%), making this result quite unusual. The key insight is that while not impossible, this outcome is rare enough to be considered unexpected, suggesting either unusual luck or that the true rate might be higher than 40%.
11. A factory reports that 2% of its light bulbs are defective. A quality-control inspector randomly checks 50 bulbs and finds 4 defective. The inspector expected about 1 defective bulb but observed 4. Is the result unexpected, given chance variation in a sample of 50?
    Answer: Yes; 4 out of 50 is unusually high if the true defect rate is 2%. The skill here is assessing whether an observed outcome is unexpected under a binomial model, aligned with AP Statistics' focus on introducing statistics and variability. With n=50 bulbs and p=0.02 defect rate, the expected defects are 1, and the standard deviation is about 0.99, so expected variability keeps most results between 0 and 3. Finding 4 defects is over 3 standard deviations above the mean, making it unexpectedly high. Distractor choice C argues it's common to see several defects, but probability calculations (around 2%) show it's rare. Mini-lesson on surprise: Use the binomial mean and SD to define a typical range (mean ± 2-3 SD); outcomes outside this are surprising because their probability is very low under the assumed model.
12. A coffee shop manager believes the average wait time for a drink during the morning rush is about 4 minutes. On one randomly selected morning, the manager records the wait times for 8 customers and finds an average of 6.5 minutes. The manager expected the sample mean to be near 4 minutes, but the observed mean was higher. Is the result unexpected, considering the variability of sample means from small random samples?
    Answer: No; with only 8 customers, the sample mean can vary noticeably from day to day, so 6.5 minutes could plausibly occur by chance. 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.
13. A game booth advertises that players win a small prize 30% of the time. A group of friends plays independently 15 times and wins 9 prizes (60%). They expected around 4 or 5 wins, but observed 9. Is the result unexpected, given the variability in the number of wins for only 15 plays if the true win probability is 30%?
    Answer: 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. 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.
14. 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?
    Answer: Yes; 5 out of 8 is unusually high if the true chance is 10%, so it would be surprising. 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.
15. A manufacturer claims that light bulbs from a certain line last an average of 1,000 hours. A consumer group randomly selects 10 bulbs and tests them, finding an average lifetime of 920 hours. The expected behavior is a sample mean near 1,000 hours, but the observed mean is lower. Is this result unexpected, considering the variability in sample means from a sample size of 10?
    Answer: No; with only 10 bulbs, the sample mean can vary, so 920 hours could plausibly occur by chance even if the true mean is 1,000 hours. This question tests understanding of sampling variability in means with small samples. Testing only 10 light bulbs and finding a mean of 920 hours versus the claimed 1,000 hours represents an 80-hour difference. With such a small sample, this deviation could easily occur by chance due to natural variation in bulb lifetimes. The distractors incorrectly suggest either that this difference is impossible or that sample means don't relate to population means. When evaluating whether a sample mean is unexpected, we must recognize that small samples (like 10 bulbs) can produce means that differ substantially from the population mean simply due to random variation, without necessarily indicating the manufacturer's claim is false.
16. A school nurse estimates that about 10% of students are absent on a typical day. On one randomly selected day, she checks a random sample of 40 students and finds 9 absent (22.5%). She wonders if this is surprising due to chance variation in a sample of 40 when the true absence rate is 10%. Is the result unexpected?
    Answer: Yes; 22.5% is unexpected because it is more than double 10% for a sample of 40. This AP Statistics question assesses the skill of judging if a sample outcome is surprising given expected variability when introducing statistics concepts. For a 10% true absence rate with n=40, the standard deviation is approximately 0.047, implying typical proportions between about 0.6% and 19.4% (within 2 SDs). The 22.5% observed is about 2.7 SDs away, indicating it's unexpected due to random sampling alone. Distractor choice A claims small samples always match closely, which is false as small n actually increases variability. Mini-lesson: To evaluate surprise, compute SD = sqrt(p(1-p)/n) and see if the difference exceeds 2-3 SDs; larger deviations suggest the result is not just chance. This helps in deciding whether to question the assumed population parameter.
17. A genetics lesson states that, for a certain trait in pea plants, each plant has a 25% chance of showing the recessive phenotype. A student grows 12 plants and observes 6 with the recessive phenotype. The student expected about 3 but observed 6. Is the result unexpected, given chance variation with only 12 plants?
    Answer: Yes; 6 out of 12 is unusually high if the true probability is 25%. This tests evaluating unexpected binomial outcomes, key in AP Statistics. n=12 plants, p=0.25, expected 3, SD 1.5, typical 0-6. 6 is exactly 2 SD above, probability ~5%, unusually high and unexpected. Distractor C says small n means not surprising, but even small n can have rare extremes. Mini-lesson: Use SD to gauge; 2+ SD away indicates surprise, occurring <5% by chance.
18. A fitness app claims that a new reminder feature increases the weekly proportion of users who complete at least 3 workouts from about 40% to about 45%. A developer tests the feature on 20 randomly selected users for one week and finds that 6 of the 20 complete at least 3 workouts (30%). This is lower than the usual 40% even though the feature was expected to help. Is this result unexpected, given variability in individual workout behavior and the sample size?
    Answer: No—because with only 20 users, the observed proportion can easily be below 40% due to random variation. 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.
19. A hospital reports that 5% of patients miss their scheduled appointment. On a particular day, 40 independently scheduled patients are expected, so the clinic expects about 2 no-shows, but observes 6 no-shows. Assuming the 5% rate is correct, is this result unexpected given day-to-day variability?
    Answer: Yes; 6 out of 40 (15%) is notably higher than 5%, so it is unusual though not impossible. This question examines whether 6 no-shows out of 40 patients is unexpected when the historical rate is 5%. With a 5% no-show rate, we expect about 2 out of 40 patients to miss appointments, so observing 6 is triple the expected number. The probability of getting 6 or more no-shows when the true rate is 0.05 is approximately 2.3%, making this an unusual occurrence. While not impossible, this result is surprising enough to suggest either an unlucky day or perhaps some external factor affecting attendance. Students should recognize that day-to-day variability exists, but some outcomes are unusual enough to warrant investigation. Understanding when results cross from "normal variability" to "unexpected" is a key statistical skill.
20. 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?
    Answer: Yes; 0 out of 20 is very low if the true rate is 30%, so it is unusual though possible. 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.

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