Justifying Claims: Confidence Interval, Population Proportion - AP Statistics

1. A streaming service claims that fewer than 25% of its subscribers watch documentaries at least once per week. From a random sample of subscribers, a 99% confidence interval for the true proportion $p$ who watch documentaries weekly is $(0.18,\ 0.24)$. Is the claim supported by the confidence interval?
   Answer: Yes, because all values in the interval are less than 0.25. This question tests evaluating a claim about a maximum proportion using a confidence interval. The claim states that fewer than 25% watch documentaries (p < 0.25), and the interval (0.18, 0.24) contains only values less than 0.25. Since every plausible value in the interval satisfies p < 0.25, the claim is supported. Choice B incorrectly focuses on the midpoint. Choice C misinterprets the confidence level. Choice D confuses the interval with a percentage of subscribers. When a confidence interval contains only values that satisfy the claim, the claim is supported by the data.
2. A public health official claims that more than 10% of adults in a county have received a flu shot this month. A random sample is used to compute an 80% confidence interval for the true proportion $p$: $(0.09,\ 0.13)$. Is the claim supported by the confidence interval?
   Answer: No, because the interval includes values less than or equal to $0.10$, so it does not support the claim $p>0.10$. This question tests whether a claim that more than 10% received flu shots (p > 0.10) is supported by an 80% CI of (0.09, 0.13). The confidence interval includes values less than or equal to 0.10 (from 0.09 to 0.10), meaning some plausible values for p do not satisfy p > 0.10. Choice B correctly identifies this - when the CI contains values that don't satisfy the inequality, the claim is not supported. Choice A misinterprets confidence as probability about p, Choice C incorrectly thinks containment proves the claim, Choice D wrongly focuses only on the upper bound, and Choice E dismisses the validity of 80% confidence. The key lesson: for strict inequality claims like p > value, the entire CI must exceed that value.
3. A wildlife biologist claims that between 30% and 40% of a certain bird species in a region carry a particular parasite. A random sample of 250 birds is tested, and a 95% confidence interval for the true proportion carrying the parasite is $(0.33, 0.39)$. Is the claim supported by the confidence interval?
   Answer: Yes, because the entire interval lies within 0.30 to 0.40, supporting the claim. This question evaluates whether a confidence interval supports a range claim. The biologist claims the proportion is BETWEEN 30% AND 40%, and the confidence interval (0.33, 0.39) falls entirely within this range. Since every plausible value for the true proportion (according to our interval) satisfies the claim, the interval supports it. Choice C incorrectly assumes the claimed values must be endpoints of the interval, and Choice D misinterprets the confidence level as a probability statement about the parameter. When a confidence interval is entirely contained within a claimed range, it provides strong support for that claim. This is different from directional claims - here we need the interval to be a subset of the claimed range.
4. A researcher claims that at most 15% of adults in a town smoke cigarettes. A random sample is taken and a 98% confidence interval for the population proportion who smoke is $(0.11, 0.15)$. Is the claim supported by the confidence interval?
   Answer: Yes, because the entire interval is at or below 0.15, supporting that the true proportion is at most 0.15. This AP Statistics question evaluates using confidence intervals to justify upper-bound claims on population proportions. The researcher claims at most 0.15 of adults smoke, and the 98% confidence interval (0.11, 0.15) is entirely <= 0.15, so all plausible values meet the claim, supporting it. Choice C is a distractor, incorrectly stating that including 0.15 means the proportion must be greater than 0.15, misapplying inclusion logic. Mini-lesson: For 'at most p,' the upper bound <= p ensures support, as no values exceed p. Here, 0.15 <= 0.15 confirms it. A is the correct answer, demonstrating how intervals can validate bounds.
5. A museum director claims that more than 35% of visitors are from out of town. Based on a random sample of visitors, an 88% confidence interval for the population proportion of out-of-town visitors is $(0.31, 0.36)$. Is the claim supported by the confidence interval?
   Answer: No, because the interval includes values less than or equal to 0.35, so it does not support that the true proportion is more than 0.35. In AP Statistics, this assesses justifying directional claims with confidence intervals for proportions. The director claims more than 0.35 are out-of-town visitors, but the 88% confidence interval (0.31, 0.36) includes values <= 0.35, so the true proportion could be 0.35 or less, not supporting the claim. Distractor B wrongly claims that including 0.35 means it's more than 0.35, confusing containment with directionality. Mini-lesson: 'More than p' requires the entire interval > p; values <= p in the interval undermine support. With 0.31 < 0.35, the claim lacks backing. Choice C is correct, highlighting precise claim-interval alignment.
6. A political analyst randomly samples 1,000 likely voters to estimate the proportion who approve of the current governor. The analyst wants to claim that the approval rate is 52% (that is, $p=0.52$). A 95% confidence interval for the population proportion approving is $(0.49, 0.55)$. Is the claim supported by the confidence interval?
   Answer: Yes, because $0.52$ is contained in the interval, so the claim $p=0.52$ is plausible. This question tests understanding of using confidence intervals to assess exact equality claims. The claim is that the approval rate equals 52% (p = 0.52), and the 95% confidence interval is (0.49, 0.55). Since 0.52 is contained within the confidence interval, it represents a plausible value for the true population proportion. When assessing equality claims, we check if the hypothesized value falls within the interval - if it does, the claim is plausible and supported by the data. The correct answer recognizes that containing 0.52 means this value is consistent with our sample data. Important distinction: for equality claims (p = k), we check if k is in the interval; for inequality claims (p > k or p < k), we check if the entire interval satisfies the inequality.
7. A nonprofit claims that at least 35% of households in a county volunteer at least once per year. From a random sample of 600 households, a 95% confidence interval for the true proportion is $(0.32, 0.36)$. Is the claim supported by the confidence interval?
   Answer: No, because the interval includes values less than 0.35, so it does not support that the true proportion is at least 0.35. This question asks whether a confidence interval supports a claim about a proportion being at least a certain value. The nonprofit claims at least 35% of households volunteer, but the 95% confidence interval (0.32, 0.36) includes values below 0.35. Since the interval contains values less than 0.35 (from 0.32 to 0.35), we cannot conclude with 95% confidence that the true proportion is at least 0.35. The distractor in choice A makes the error of thinking that because 0.35 is in the interval, the claim is supported—but for "at least" claims, we need all values in the interval to meet the criterion. When using confidence intervals to evaluate claims with inequality statements, the entire interval must satisfy the inequality to support the claim.
8. A university claims that at most 30% of its undergraduates have taken an online course. A random sample was used to create a 95% confidence interval for the true proportion $p$ who have taken an online course: $(0.28,\ 0.34)$. Is the claim supported by the confidence interval?
   Answer: No, because the interval includes values greater than 0.30, so it does not support $p\le 0.30$. This question asks about evaluating a maximum claim using a confidence interval. The claim states that at most 30% have taken online courses (p ≤ 0.30), but the interval (0.28, 0.34) includes values above 0.30. Since some plausible values of p exceed 0.30, the claim is not supported. Choice A incorrectly thinks containing 0.30 proves the claim. Choice C misinterprets what the confidence level means. Choice E wrongly focuses only on the lower bound. To support a claim that p ≤ some value, the entire confidence interval must be at or below that value.
9. A candidate claims that exactly 50% of voters in a district support her. A random sample of 1,000 voters yields a 95% confidence interval for the true support proportion of $(0.47, 0.51)$. Is the claim supported by the confidence interval?
   Answer: Yes, because 0.50 is contained in the interval, so the claim that the true proportion is 0.50 is plausible. This question examines whether a confidence interval can support a claim about an exact value for a population proportion. The candidate claims exactly 50% support, and the 95% confidence interval (0.47, 0.51) contains 0.50. Since 0.50 is within the interval, it is a plausible value for the true proportion, so the claim is supported by the data. The distractor in choice C incorrectly suggests that because the interval contains other values, the true proportion cannot be 0.50—but confidence intervals show a range of plausible values, not just one exact value. When evaluating claims about exact values, we check if that value falls within the confidence interval; if it does, the claim is plausible and supported by the data.
10. A university randomly samples 900 undergraduates to estimate the proportion who have taken at least one online course. The university wants to claim that more than 30% of undergraduates have taken an online course. A 98% confidence interval for the population proportion is $(0.29, 0.34)$. Is the claim supported by the confidence interval?
    Answer: No, because $0.30$ is inside the interval, so the data do not provide convincing evidence that $p>0.30$. This question tests whether a confidence interval can justify a claim about a proportion exceeding a specific value. The claim is that more than 30% have taken an online course (p > 0.30), and the 98% confidence interval is (0.29, 0.34). Since 0.30 is contained within the confidence interval, it represents a plausible value for the true population proportion. When a confidence interval contains a hypothesized value, we cannot reject that value as a possibility, which means we cannot claim the proportion is definitively greater than it. The correct answer recognizes that containing 0.30 means we lack convincing evidence that the true proportion exceeds 0.30. Key insight: the presence of 0.30 in the interval means p = 0.30 is plausible, contradicting the claim p > 0.30.
11. A public health official claims that fewer than 10% of adults in a region smoke cigarettes. A random sample was used to compute a 98% confidence interval for the true proportion $p$ of adults who smoke: $(0.09,\ 0.12)$. Is the claim supported by the confidence interval?
    Answer: No, because the interval includes values greater than 0.10, so it does not support $p<0.10$. This question tests evaluating a maximum claim using a confidence interval. The claim is that fewer than 10% smoke (p < 0.10), but the interval (0.09, 0.12) includes values above 0.10. Since some plausible values of p are greater than or equal to 0.10, the claim is not supported. Choice A incorrectly thinks containing 0.10 would support p < 0.10. Choice C misinterprets the confidence level. Choice E wrongly focuses only on the lower bound. A confidence interval supports a claim only when all values in the interval satisfy the claim's condition.
12. A manufacturer claims that at least 90% of customers are satisfied with a product. A random sample of customers yields an 80% confidence interval for the true proportion $p$ satisfied: $(0.88,\ 0.93)$. Is the claim supported by the confidence interval?
    Answer: No, because the interval includes values below 0.90, so it does not support $p\ge 0.90$. This question tests evaluating a minimum claim with a confidence interval. The claim is that at least 90% are satisfied (p ≥ 0.90), but the interval (0.88, 0.93) includes values below 0.90. Since the interval contains values less than 0.90, we cannot conclude that p ≥ 0.90. Choice A incorrectly assumes containing 0.90 proves the claim. Choice C misinterprets the confidence level. Choice D wrongly focuses only on the upper bound. When a confidence interval includes values that contradict the claim, the claim is not supported, regardless of the confidence level used.
13. A museum randomly samples 650 visitors to estimate the proportion who would recommend the museum to a friend. The museum wants to claim that at most 70% of all visitors would recommend it (that is, $p\le 0.70$). A 90% confidence interval for the population proportion is $(0.68, 0.74)$. Is the claim supported by the confidence interval?
    Answer: No, because the interval includes values greater than $0.70$, so it does not support the claim $p\le 0.70$. This question examines whether a confidence interval supports a claim about a proportion being at most a certain value. The claim is that at most 70% would recommend (p ≤ 0.70), and the 90% confidence interval is (0.68, 0.74). Since the interval includes values greater than 0.70 (specifically 0.71 through 0.74), we cannot conclude that the true proportion is definitely at most 0.70. The interval suggests the true proportion could plausibly be anywhere from 68% to 74%, which includes possibilities where p > 0.70. The correct answer recognizes this principle: to support p ≤ k, the entire interval must be at or below k. When the interval extends above k, the claim is not supported by the data.
14. A school principal claims that fewer than 40% of students prefer block scheduling. A random sample of students is surveyed, and a 95% confidence interval for the true proportion $p$ who prefer block scheduling is $(0.33,\ 0.39)$. Is the claim supported by the confidence interval?
    Answer: Yes, because every value in the interval is less than $0.40$, supporting the claim $p<0.40$. This question asks whether a claim that fewer than 40% prefer block scheduling (p < 0.40) is supported by a 95% CI of (0.33, 0.39). Since every value in the confidence interval is less than 0.40, all plausible values for p satisfy p < 0.40. Choice A correctly identifies this - when the entire CI satisfies the inequality in the claim, the claim is supported. Choice B misinterprets the confidence level, Choice C incorrectly thinks non-containment prevents evaluation, Choice D misunderstands what non-containment means, and Choice E wrongly dismisses the validity of 95% confidence. The key principle: a claim is supported when ALL values in the CI make the claim true.
15. A nutrition label company claims that at least 25% of adults regularly read nutrition labels. A random sample is taken and a 90% confidence interval for the population proportion $p$ is computed as $(0.27,\ 0.34)$. Is the claim supported by the confidence interval?
    Answer: Yes, because all values in the interval are at least $0.25$, so the claim $p\ge 0.25$ is supported. This question asks whether a claim that at least 25% of adults read nutrition labels (p ≥ 0.25) is supported by a 90% CI of (0.27, 0.34). The key insight is that every value in the confidence interval is greater than 0.25, which means all plausible values for p satisfy p ≥ 0.25. Choice B correctly identifies this - when all values in the CI satisfy the claim, the claim is supported. Choice A misinterprets the confidence level, Choice C irrelevantly focuses on the midpoint, Choice D confuses sample statistics with the confidence level, and Choice E misunderstands that we're testing p ≥ 0.25, not p = 0.34. When using CIs for hypothesis testing, check if all values in the interval satisfy the claim's inequality.
16. A smartphone manufacturer claims that at least 95% of its phones last one full year without needing a repair. From a random sample of customers, a 90% confidence interval for the population proportion of phones lasting one year without repair is $(0.94, 0.98)$. Is the claim supported by the confidence interval?
    Answer: No, because the interval includes values below 0.95, so it does not support that the true proportion is at least 0.95. Justifying claims with confidence intervals for proportions is the AP Statistics skill here. The manufacturer claims at least 0.95 of phones last a year, but the 90% confidence interval (0.94, 0.98) includes values below 0.95, indicating the true proportion could be less, so the claim isn't supported. Distractor E suggests the upper bound exceeding 0.95 confirms it, but this ignores the lower bound's role. Mini-lesson: Claims like 'at least p' require the lower bound >= p for support, ensuring no plausible values contradict it. Since 0.94 < 0.95, support is lacking. Choice B is correct, emphasizing comprehensive interval evaluation.
17. A city council surveys a random sample of 450 registered voters about whether they support a proposed public transit tax. The council plans to claim that more than 60% of all registered voters support the tax. A 95% confidence interval for the population proportion of supporters is $(0.57, 0.63)$. Is the claim supported by the confidence interval?
    Answer: No, because $0.60$ is contained in the interval, so the data do not provide convincing evidence that the true proportion is greater than $0.60$. This question tests whether you can use a confidence interval to justify a claim about a population proportion being greater than a specific value. The claim is that more than 60% support the tax (p > 0.60), and the 95% confidence interval is (0.57, 0.63). Since 0.60 is contained within the confidence interval, it represents a plausible value for the true population proportion. When a confidence interval contains a hypothesized value, we cannot reject that value, which means we cannot claim the proportion is definitively greater than it. The correct answer recognizes that containing 0.60 means we lack convincing evidence that the true proportion exceeds 0.60. Remember: to support a claim that p > some value, the entire confidence interval must be above that value.
18. A city planner claims that more than 60% of residents support a new bike-lane proposal. A random sample of residents is used to construct a 95% confidence interval for the true proportion $p$ of all residents who support the proposal: $(0.57,\ 0.63)$. Is the claim supported by the confidence interval?
    Answer: No, because $0.60$ is in the interval, so the data do not provide convincing evidence that $p>0.60$. This question tests understanding of using confidence intervals to evaluate claims about population proportions. The claim states that more than 60% support the proposal (p > 0.60), and we have a 95% CI of (0.57, 0.63). Since the confidence interval contains 0.60, we cannot conclude that p > 0.60 with confidence. Choice B correctly identifies this - when the hypothesized value falls within the CI, we do not have convincing evidence against it. Choice A misinterprets what the 95% means (it's about confidence in the interval, not about the percentage of residents), Choice C incorrectly assumes that having values above 0.60 proves the claim, Choice D wrongly interprets containment as proving an exact value, and Choice E confuses the confidence level with a probability statement about p.
19. A nonprofit claims that fewer than 15% of donors give anonymously. A random sample of 350 donors is selected, and a 95% confidence interval for the true proportion of anonymous donors is $(0.14, 0.19)$. Is the claim supported by the confidence interval?
    Answer: No, because the interval includes values greater than 0.15, so the data do not support the claim that the true proportion is fewer than 15%. This question tests understanding of using confidence intervals to evaluate "fewer than" claims. The nonprofit claims that FEWER THAN 15% of donors give anonymously, meaning the true proportion is less than 0.15. The confidence interval (0.14, 0.19) includes values above 0.15, such as 0.16, 0.17, and 0.19, which means proportions greater than 0.15 are plausible. Since we cannot rule out proportions at or above 0.15, we cannot support the claim. Choice A incorrectly interprets the presence of 0.15 in the interval as support for the claim, and Choice E focuses only on the lower bound. For "fewer than" claims to be supported, the entire confidence interval must be below the claimed value.
20. A city council member claims that more than 60% of residents support a proposed public transit expansion. In a random sample of residents, a 95% confidence interval for the population proportion who support the expansion is $(0.56, 0.64)$. Is the claim supported by the confidence interval?
    Answer: No, because the interval includes values less than 0.60, so it does not provide evidence that the true proportion is more than 0.60. This question assesses the skill of justifying claims about a population proportion using a confidence interval in AP Statistics. The claim is that more than 0.60 of residents support the expansion, but the 95% confidence interval (0.56, 0.64) includes values below 0.60, meaning the true proportion could plausibly be less than or equal to 0.60, so the interval does not support the claim. A common distractor is choice A, which incorrectly assumes that since 95% of residents are in the interval and the upper end is above 0.60, the claim is supported, but this misinterprets the confidence level and interval containment. In a mini-lesson on using confidence intervals for claims: a confidence interval provides a range of plausible values for the true population proportion; for a claim like 'more than p,' the entire interval must be above p to support it, as any value below p in the interval suggests the claim might not hold. Here, since the lower bound is 0.56 < 0.60, the claim is not supported, making choice C correct. Remember, the confidence level indicates the reliability of the interval construction method, not the probability for a specific interval.

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