AP Statistics Quiz: Justifying Claims Confidence Interval Population Proportion

What this quiz covers

This quiz focuses on Justifying Claims Confidence Interval Population Proportion, 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. No, because 95% confidence means 95% of subscribers use the service daily.
2. Yes, because the entire confidence interval lies within $[0.45,0.55]$, making the claim plausible. (correct answer)
3. No, because the interval does not include $0.55$.
4. Yes, because there is a 95% probability that $p$ is exactly $0.49$.
5. No, because the interval includes more than one value, so no range claim can be supported.

Explanation: This question examines whether a confidence interval supports a range claim about a population proportion. The claim is that between 45% and 55% use the service daily (0.45 ≤ p ≤ 0.55), and the 95% confidence interval is (0.46, 0.52). Since the entire confidence interval (0.46, 0.52) lies within the claimed range [0.45, 0.55], all plausible values for the true proportion satisfy the claim. This provides support because every value in our confidence interval falls within the proposed range. The correct answer properly identifies that when a confidence interval is entirely contained within a claimed range, it supports that range claim. This is a special case where we're checking if the CI is a subset of the claimed interval.

Question 2

1. Yes, because 0.80 is in the interval, so $p\ge 0.80$ must be true.
2. No, because the interval includes values below 0.80, so it does not support $p\ge 0.80$. (correct answer)
3. Yes, because 90% confidence means there is a 90% chance $p\ge 0.80$.
4. No, because 90% of students are between 0.78 and 0.85.
5. Yes, because the upper end of the interval is above 0.80.

Explanation: This question asks whether a confidence interval supports a claim about a minimum proportion. The claim is that at least 80% eat breakfast (p ≥ 0.80), but the interval (0.78, 0.85) includes values below 0.80. Since some plausible values of p are less than 0.80, we cannot conclude that p ≥ 0.80. Choice A incorrectly thinks containing 0.80 proves the claim. Choice C misinterprets the confidence level. Choice D confuses the confidence interval with a range of student percentages. To support a claim that p ≥ some value, the entire confidence interval must be at or above that value.

Question 3

1. Yes, because 92% confidence means 92% of readers prefer Android.
2. No, because $0.35$ is in the interval, so the data do not provide convincing evidence that $p<0.35$. (correct answer)
3. Yes, because the lower endpoint is below $0.35$.
4. No, because a confidence interval cannot be used to assess a claim about being less than a value.
5. Yes, because there is a 92% probability that the true proportion is between $0.33$ and $0.38$, and $0.35$ is closer to $0.33$ than to $0.38$.

Explanation: This question examines whether a confidence interval can justify a claim about a proportion being less than a specific value. The claim is that fewer than 35% prefer Android (p < 0.35), and the 92% confidence interval is (0.33, 0.38). Since 0.35 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 less than it. The correct answer recognizes that containing 0.35 means we lack convincing evidence that the true proportion is less than 0.35. Remember: to support p < k, the entire interval must be below k; if k is in the interval, p = k is plausible.

Question 4

1. Yes, because 95% of residents are in the interval.
2. Yes, because 0.60 is in the interval, so $p>0.60$.
3. No, because the interval includes values less than 0.60, so it does not support $p>0.60$. (correct answer)
4. No, because 95% confidence means there is a 95% chance that $p$ is less than 0.60.
5. Yes, because the sample proportion must be 0.60 exactly.

Explanation: This question tests whether you can use a confidence interval to evaluate a claim about a population proportion. The claim states that more than 60% support the ordinance (p > 0.60), but the confidence interval (0.56, 0.64) includes values both above and below 0.60. Since the interval contains values less than 0.60, we cannot conclude that p > 0.60. Choice A incorrectly interprets the confidence level as referring to the percentage of residents. Choice B wrongly assumes that containing 0.60 supports p > 0.60. Choice D misinterprets the confidence level as a probability about p. When using confidence intervals to test claims, if the interval includes values that contradict the claim, the claim is not supported.

Question 5

1. Yes, because the interval includes values above $0.65$, so it proves $p\ge 0.65$.
2. No, because the interval includes values below $0.65$, so it does not support the claim $p\ge 0.65$. (correct answer)
3. Yes, because 95% confidence means there is a 95% chance that $p\ge 0.65$.
4. No, because $0.65$ is not the upper endpoint of the interval.
5. Yes, because the midpoint of the interval is close to $0.65$.

Explanation: This question tests understanding of using confidence intervals to support "at least" claims. The claim is that at least 65% received a flu shot (p ≥ 0.65), and the 95% confidence interval is (0.62, 0.66). Since the interval includes values below 0.65 (specifically 0.62 through 0.64), we cannot conclude that the true proportion is definitely at least 0.65. The interval tells us the true proportion could plausibly be anywhere from 62% to 66%, which includes possibilities where p < 0.65. The correct answer recognizes this principle: to support p ≥ k, the entire interval must be at or above k. When the interval contains values below k, the claim is not supported by the data.

Question 6

1. Yes, because 0.35 is in the interval.
2. Yes, because 95% confidence means there is a 95% chance $p>0.35$.
3. No, because the interval includes values less than or equal to 0.35, so it does not support $p>0.35$. (correct answer)
4. No, because 95% of households compost.
5. Yes, because the upper bound is greater than 0.35.

Explanation: This question asks whether a confidence interval supports a claim about exceeding a proportion. The claim states that more than 35% compost (p > 0.35), but the interval (0.31, 0.36) includes values both above and below 0.35, including 0.35 itself. Since the interval contains values less than or equal to 0.35, the claim is not supported. Choice A incorrectly thinks containing 0.35 supports p > 0.35. Choice B misinterprets confidence levels. Choice E wrongly focuses only on the upper bound. To support a strict inequality claim (p > value), the entire interval must exceed that value.

Question 7

1. Yes, because the interval contains $0.70$, so it proves $p\le 0.70$.
2. No, because the interval includes values greater than $0.70$, so it does not support the claim $p\le 0.70$. (correct answer)
3. Yes, because 90% confidence means 90% of visitors would recommend the museum.
4. No, because $0.70$ is not the midpoint of the interval.
5. Yes, because the lower bound is below $0.70$.

Explanation: 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.

Question 8

1. Yes, because the interval's upper bound is above 0.35, so the true proportion must be more than 0.35.
2. Yes, because 0.35 is inside the interval, so the true proportion is more than 0.35.
3. 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. (correct answer)
4. No, because 88% confidence means there is an 88% probability the true proportion is less than 0.35.
5. Yes, because an 88% confidence interval is wide enough to confirm the claim.

Explanation: 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.

Question 9

1. No, because 95% confidence means there is a 95% probability that $p$ is between 0.35 and 0.45.
2. Yes, because the entire confidence interval lies within $(0.35, 0.45)$, so the claim is supported. (correct answer)
3. No, because the interval does not include exactly $0.35$ and $0.45$ as endpoints.
4. Yes, because 95% of customers in the sample bought a magazine.
5. No, because the interval contains 0.40, which proves $p=0.40$ exactly rather than a range.

Explanation: This question asks whether a claim that between 35% and 45% buy magazines (0.35 < p < 0.45) is supported by a 95% CI of (0.37, 0.44). Since the entire confidence interval (0.37, 0.44) lies within the claimed range (0.35, 0.45), all plausible values for p satisfy the claim. Choice B correctly identifies this - when the CI is entirely contained within the claimed range, the claim is supported. Choice A misinterprets what confidence means, Choice C incorrectly requires exact endpoint matching, Choice D confuses sample statistics with confidence levels, and Choice E wrongly interprets a contained value as proving exact equality. For range claims, check if the entire CI falls within the specified range.

Question 10

1. Yes, because the interval contains values above $0.70$, so $p\ge 0.70$ is guaranteed.
2. No, because the interval includes values less than $0.70$, so the data do not support the claim $p\ge 0.70$. (correct answer)
3. Yes, because 92% confidence means 92% of subscribers watch weekly.
4. Yes, because there is a 92% probability that $p$ is at least $0.70$.
5. No, because $0.70$ is not the midpoint of the interval, so the claim is false.

Explanation: This question tests whether a claim that at least 70% watch weekly (p ≥ 0.70) is supported by a 92% CI of (0.68, 0.74). The confidence interval includes values less than 0.70 (specifically 0.68 to 0.70), meaning some plausible values for p do not satisfy p ≥ 0.70. Choice B correctly identifies this - when the CI contains values that violate the claim, we cannot support it. Choice A wrongly focuses only on values above 0.70, Choice C confuses the confidence level with the proportion, Choice D misinterprets confidence as probability about p, and Choice E irrelevantly discusses the midpoint. Remember: for claims with inequalities, check whether ALL values in the CI satisfy the inequality.

Question 11

1. No, because 95% confidence means the candidate has only a 95% chance of winning.
2. Yes, because all values in the interval are greater than $0.52$, supporting the claim $p>0.52$. (correct answer)
3. No, because $0.52$ is not included in the interval, so the claim cannot be tested.
4. Yes, because the interval contains $0.58$, which proves support is 58%.
5. No, because there is a 5% probability that $p$ is outside the interval, so the claim is unsupported.

Explanation: This question asks whether a claim that more than 52% support the candidate (p > 0.52) is supported by a 95% CI of (0.53, 0.58). Since every value in the confidence interval is greater than 0.52, all plausible values for p satisfy p > 0.52. Choice B correctly identifies this - when all values in the CI satisfy the claim's inequality, the claim is supported. Choice A misinterprets the confidence level, Choice C incorrectly thinks non-inclusion prevents testing, Choice D wrongly interprets a single value as proof of an exact proportion, and Choice E confuses the interpretation of confidence levels. The principle for using CIs with inequality claims: the claim is supported when the entire interval satisfies the inequality.

Question 12

1. No, because 0.50 is not in the interval.
2. Yes, because 0.50 is contained in the interval, so the data are consistent with $p=0.50$. (correct answer)
3. Yes, because 95% confidence means there is a 95% chance $p=0.50$.
4. No, because the interval includes more than one value, so $p$ cannot be 0.50.
5. No, because 95% of customers have takeout proportions between 0.47 and 0.53.

Explanation: This question asks about testing an exact claim using a confidence interval. The claim is that exactly 50% order takeout (p = 0.50), and the interval (0.47, 0.53) contains 0.50. Since 0.50 is a plausible value for p based on the data, the claim is consistent with the confidence interval. Choice A incorrectly states 0.50 is not in the interval. Choice C misinterprets the confidence level. Choice D wrongly thinks an interval can't support a specific value. Choice E confuses the interval with customer percentages. A confidence interval supports a claim about a specific value if that value is contained within the interval.

Question 13

1. Yes, because the upper end of the interval is greater than $0.30$.
2. No, because 98% confidence means there is a 98% chance that the sample proportion is between $0.29$ and $0.34$.
3. No, because $0.30$ is inside the interval, so the data do not provide convincing evidence that $p>0.30$. (correct answer)
4. Yes, because 98% of undergraduates have taken an online course.
5. Yes, because the lower bound is close to $0.30$, which guarantees $p>0.30$.

Explanation: 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.

Question 14

1. No, because 0.48 is not in the interval.
2. Yes, because 90% confidence means there is a 90% chance $p>0.48$.
3. Yes, because the entire interval is above 0.48, supporting $p>0.48$. (correct answer)
4. No, because the interval contains 0.50, so $p$ must equal 0.50.
5. No, because 90% of likely voters are between 0.49 and 0.55.

Explanation: This question tests whether a confidence interval supports a claim about a minimum proportion. The claim is that more than 48% support the candidate (p > 0.48), and the entire interval (0.49, 0.55) is above 0.48. Since all plausible values of p exceed 0.48, the claim is strongly supported. Choice A incorrectly states 0.48 is not in the interval (which would be irrelevant anyway). Choice B misinterprets the confidence level. Choice D makes an unfounded claim about p equaling 0.50. When every value in a confidence interval satisfies the claim, the claim is supported by the data.

Question 15

1. Yes, because 0.10 is in the interval.
2. No, because the interval includes values greater than 0.10, so it does not support $p<0.10$. (correct answer)
3. Yes, because 98% confidence means there is a 98% chance that $p<0.10$.
4. No, because 98% of adults have smoking rates between 0.09 and 0.12.
5. Yes, because the lower bound is below 0.10.

Explanation: 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.

Question 16

1. Yes, because 95% confidence means there is a 95% chance $p=0.40$.
2. No, because 0.40 is not in the interval, so the data do not support $p$ being close to 0.40. (correct answer)
3. Yes, because the interval is narrow, so any claim is supported.
4. Yes, because 95% of users are between 0.41 and 0.47.
5. No, because 0.40 is the midpoint of the interval.

Explanation: This question asks about testing whether a proportion is "about" a specific value using a confidence interval. The claim suggests p is close to 0.40, but the interval (0.41, 0.47) does not contain 0.40. Since 0.40 is outside the interval, it's not a plausible value for p based on the data, so the claim is not supported. Choice A misinterprets confidence levels. Choice C incorrectly suggests narrow intervals support any claim. Choice E incorrectly calculates the midpoint (which would be 0.44). When testing if p is near a specific value, that value should be within the confidence interval to support the claim.

Question 17

1. Yes, because 0.90 is in the interval, so $p\ge 0.90$ is confirmed.
2. No, because the interval includes values below 0.90, so it does not support $p\ge 0.90$. (correct answer)
3. Yes, because 80% confidence means there is an 80% probability that $p\ge 0.90$.
4. Yes, because the upper end is above 0.90.
5. No, because 80% of customers are between 0.88 and 0.93.

Explanation: 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.

Question 18

1. No, because 90% confidence means there is a 10% chance the claim is false.
2. Yes, because all values in the interval are at least $0.25$, so the claim $p\ge 0.25$ is supported. (correct answer)
3. No, because $0.25$ is not the midpoint of the interval.
4. Yes, because 90% of adults in the sample read labels.
5. No, because the interval does not contain $p=0.34$ exactly.

Explanation: 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.

Question 19

1. Yes, because 99% confidence means there is a 99% probability that $p\le 0.08$.
2. No, because the interval includes values greater than $0.08$, so the data do not support the claim $p\le 0.08$. (correct answer)
3. Yes, because $0.08$ is inside the interval, so the claim must be true.
4. No, because 99% of packages are late when using a 99% confidence interval.
5. Yes, because the lower bound is less than $0.08$, which proves $p\le 0.08$.

Explanation: This question tests whether a claim that no more than 8% of packages arrive late (p ≤ 0.08) is supported by a 99% CI of (0.06, 0.10). Since the confidence interval includes values greater than 0.08 (specifically, values from 0.08 to 0.10), we cannot conclude that p ≤ 0.08. Choice B correctly identifies this - when the CI contains values that violate the claim, the data do not support it. Choice A misinterprets the confidence level as a probability about p, Choice C incorrectly thinks containment proves the claim, Choice D confuses the confidence level with the proportion, and Choice E wrongly focuses only on the lower bound. Remember: for a claim to be supported, ALL values in the CI must satisfy the inequality.

Question 20

1. Yes, because 0.95 is within the interval, so the true proportion is at least 0.95.
2. No, because the interval includes values below 0.95, so it does not support that the true proportion is at least 0.95. (correct answer)
3. Yes, because 90% confidence means there is a 90% chance the true proportion is at least 0.95.
4. No, because a 90% confidence interval cannot be used to assess any claim involving 95%.
5. Yes, because the upper bound exceeds 0.95, which confirms the claim.

Explanation: 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.