AP Statistics Quiz: Confidence Interval For A Population Proportion

What this quiz covers

This quiz focuses on Confidence Interval For A 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. About 95% of all households in the town have a pet dog.
2. There is a 95% probability that the interval $(0.19, 0.41)$ contains the true proportion $p$.
3. We are 95% confident that the true proportion $p$ of all households in the town with a pet dog is between 0.19 and 0.41. (correct answer)
4. If we repeatedly sample 60 households, 95% of those samples will have exactly 30% with a pet dog.
5. The true proportion $p$ changes from sample to sample, and 95% of the time it will be between 0.19 and 0.41.

Explanation: The skill here is interpreting confidence intervals for population proportions in AP Statistics. The correct interpretation is that we are 95% confident the true proportion p of households with a pet dog is between 0.19 and 0.41, based on the method's 95% success rate in repeated sampling. Choice B is a distractor, incorrectly phrasing it as a 95% probability that the specific interval contains p, but since p is fixed and the interval is computed, it's not a probability for this instance but for the process. In a mini-lesson, confidence intervals reflect that over many identical studies, 95% of the calculated intervals would include the true parameter, providing a way to quantify uncertainty without assigning post-data probabilities to p. This understanding clarifies why we say 'confident' rather than 'probable' for a given interval. It also helps differentiate between the sample statistic and the population parameter.

Question 2

1. If many samples of 500 voters are taken and a 90% confidence interval is computed each time, about 90% of those intervals will contain the true proportion $p$. (correct answer)
2. There is a 90% probability that $p$ is between 0.52 and 0.58.
3. About 90% of registered voters in the city support the tax.
4. About 90% of samples of 500 voters will have sample proportion between 0.52 and 0.58.
5. Because the confidence level is 90%, the interval $(0.52, 0.58)$ contains exactly 90% of the population values.

Explanation: This question assesses the interpretation of a confidence interval for a population proportion, a key concept in AP Statistics. The correct choice states that if many samples of 500 voters are taken and 90% confidence intervals computed each time, about 90% of those intervals will contain the true proportion p, emphasizing the long-run frequency interpretation. Choice D is a distractor, suggesting that 90% of samples will have a sample proportion between 0.52 and 0.58, but this is incorrect because it confuses the interval for p with the distribution of sample proportions around the true p, not this fixed interval. A mini-lesson on confidence intervals: they provide a range where we expect the true parameter to lie, based on the idea that the sampling method produces intervals that cover the parameter 90% of the time in repeated use. This frequency approach underscores that confidence is about the reliability of the procedure, not a probability for this particular interval or the parameter itself. Mastering this helps in distinguishing between the variability of samples and the fixed population parameter.

Question 3

1. We are 96% confident that the true proportion $p$ of all customers at that store who use self-checkout is between 0.18 and 0.28. (correct answer)
2. There is a 96% chance that the true proportion $p$ is between 0.18 and 0.28.
3. About 96% of customers at that store use self-checkout.
4. If a new sample of 200 customers is taken, the interval $(0.18, 0.28)$ will be produced 96% of the time.
5. Because 96% is close to 100%, the interval must contain the true proportion $p$.

Explanation: The skill being assessed is the interpretation of a confidence interval for a population proportion in AP Statistics. Accurately, we are 96% confident the true p of self-checkout users is between 0.18 and 0.28, based on the method's reliability. Choice B distracts by stating a 96% chance p is in the interval, but this wrongly assigns probability to the fixed p rather than the random interval. In a mini-lesson, a 96% CI implies that in repeated sampling, 96% of such intervals would include the true parameter, measuring procedural confidence. This avoids errors in treating CIs as predictive probabilities. It fosters better comprehension of inferential statistics.

Question 4

1. There is a 94% chance that the true proportion $p$ is between 0.54 and 0.66.
2. We are 94% confident that between 54% and 66% of all passengers on that rail line purchase tickets using a mobile app. (correct answer)
3. 94% of passengers on that rail line purchase tickets using a mobile app.
4. In repeated sampling, 94% of samples of 350 passengers will have sample proportion between 0.54 and 0.66.
5. The probability that $p$ is in $(0.54, 0.66)$ is 0.94.

Explanation: This question in AP Statistics focuses on interpreting a confidence interval for a population proportion. The correct interpretation is that we are 94% confident between 54% and 66% of passengers use a mobile app, meaning the interval likely contains p with 94% confidence in the process. Distractor D claims 94% of repeated samples will have proportions between 0.54 and 0.66, which is false as it ignores that intervals shift with each sample's hat{p}. Mini-lesson: Confidence intervals at 94% level mean the method will enclose the true parameter 94% of the time over infinite trials, offering a structured way to express uncertainty. This interpretation prevents conflating sample statistics with population truths. It aids in effective statistical analysis and reporting.

Question 5

1. There is a 95% probability that the true proportion $p$ is between 0.39 and 0.47.
2. We are 95% confident that the interval from 0.39 to 0.47 captures the true proportion $p$ of all high school students in the district who get at least 8 hours of sleep. (correct answer)
3. About 95% of all high school students in the district get at least 8 hours of sleep.
4. If we repeated the survey many times, 95% of the students in each sample would say "yes."
5. Because the confidence level is 95%, the true proportion $p$ must be between 0.39 and 0.47.

Explanation: This question tests the skill of interpreting a confidence interval for a population proportion in AP Statistics. The correct interpretation is that we are 95% confident the interval from 0.39 to 0.47 captures the true proportion p of all high school students in the district who get at least 8 hours of sleep, as stated in choice B. A common distractor is choice A, which incorrectly treats the confidence level as a probability that the true p falls in the interval, but once the interval is calculated, p is either in it or not. In a mini-lesson on confidence intervals, remember that a 95% confidence interval means that if we repeated the sampling process many times and constructed intervals each time, about 95% of those intervals would contain the true population proportion. This confidence refers to the reliability of the method, not to a single interval or the parameter itself. Choice C mistakenly applies the 95% to the population directly, while D confuses it with sample outcomes, and E implies certainty, which isn't accurate.

Question 6

1. There is a 97% chance that the true proportion $p$ is between 0.43 and 0.49.
2. We are 97% confident that between 43% and 49% of the sampled employees primarily work remotely.
3. If many 97% confidence intervals were constructed from many random samples of 1,500 employees, about 97% of those intervals would contain the true proportion $p$. (correct answer)
4. 97% of all employees worldwide primarily work remotely.
5. The confidence interval means the true proportion $p$ will be between 0.43 and 0.49 for 97% of the employees.

Explanation: In AP Statistics, this question assesses confidence interval interpretation for remote work proportion. Choice C correctly explains that if many 97% intervals are constructed from repeated samples, about 97% would contain the true p. A common distractor is choice A, using 'chance' for p in the interval, but confidence isn't probability for the parameter. Mini-lesson: a confidence interval's level indicates the long-run proportion of intervals that capture the fixed population proportion across many samples. Choice B limits to the sample, D applies 97% to the population, and E misinterprets as p varying for individuals.

Question 7

1. We are 96% confident that the true proportion $p$ of all subscribers who watched the series in its first week is between 0.30 and 0.36. (correct answer)
2. There is a 96% probability that the true proportion $p$ is between 0.30 and 0.36.
3. If we sample again, 96% of the time the sample proportion will be exactly 0.33, the midpoint of the interval.
4. Because the confidence interval is 0.30 to 0.36, exactly 30% to 36% of the 500 sampled subscribers watched the series.
5. About 96% of subscribers watched the series in its first week.

Explanation: This question assesses confidence interval interpretation for streaming viewership. The sample proportion is 165/500 = 0.33, with a 96% CI of (0.30, 0.36). The correct answer (A) properly states we are 96% confident the true proportion of all subscribers who watched is between 0.30 and 0.36. Choice B incorrectly treats the interval as a probability statement about the parameter. Choice C misunderstands sampling variability and the meaning of the interval. Choice D is incorrect - we know exactly 33% of the sample watched, not 30-36%. Choice E confuses the confidence level with the viewership rate. A confidence interval provides a range estimate for an unknown population parameter, with the confidence level indicating the reliability of the interval construction method.

Question 8

1. We are 95% confident that the interval from 0.58 to 0.64 contains the true proportion $p$ of all adults in the state who approve. (correct answer)
2. There is a 95% chance that the true proportion $p$ equals 0.61.
3. 95% of all adults in the state approve of the governor.
4. Because the confidence interval is narrow, it must contain the true proportion $p$.
5. If the sampling were repeated, 95% of the time the sample proportion would be between 0.58 and 0.64 exactly.

Explanation: Interpreting confidence intervals for population proportions is the core skill in this AP Statistics question. The proper interpretation is that we are 95% confident the interval from 0.58 to 0.64 contains the true p of approving adults, grounded in the method's 95% capture rate. Choice E is a distractor, claiming 95% of repeated sample proportions would be exactly between 0.58 and 0.64, but this misrepresents that sample proportions vary around p, not this fixed interval. In a mini-lesson, confidence intervals indicate that repeating the sampling and interval calculation would cover the true parameter 95% of the time, quantifying estimation reliability. This frequency perspective prevents confusing sample variability with parameter certainty. It promotes precise inference in statistics.

Question 9

1. If the sampling method is repeated many times, about 92% of the confidence intervals constructed this way will contain the true proportion $p$. (correct answer)
2. There is a 92% probability that the sample proportion is between 0.55 and 0.65.
3. About 92% of all undergraduates have taken an online course.
4. We can be 92% confident that 92% of students are between 0.55 and 0.65 likely to have taken an online course.
5. There is a 92% chance that $p$ will change to be inside $(0.55, 0.65)$ if we sample again.

Explanation: This question tests understanding of confidence interval interpretation with an unusual confidence level. The sample proportion is 210/350 = 0.60, with a 92% CI of (0.55, 0.65). The correct answer (A) properly interprets the 92% confidence level as the long-run proportion of intervals that would contain the true parameter if the sampling process were repeated many times. Choice B incorrectly applies the confidence level to the sample proportion, which is known exactly. Choice C confuses the confidence level with the actual proportion of students. Choice E misunderstands the nature of the parameter - it's fixed, not changing between samples. The confidence level describes the reliability of the interval construction method, not any specific probability about this particular interval.

Question 10

1. If many random samples of 150 households were taken and a 92% confidence interval computed each time, about 92% of those intervals would contain the true proportion $p$ of households in the town that regularly recycle. (correct answer)
2. There is a 92% chance that $p$ is between 0.54 and 0.70.
3. About 92% of households in the town regularly recycle.
4. In 92% of all towns, the proportion that regularly recycle is between 0.54 and 0.70.
5. If the same 150 households were surveyed again, 92% of the time the sample proportion would be between 0.54 and 0.70.

Explanation: This question assesses understanding of confidence level interpretation. The correct answer (A) properly describes what 92% confidence means - if we took many samples and computed confidence intervals each time, about 92% of those intervals would contain the true proportion p. Choice B incorrectly treats confidence as probability about the parameter. Choice C misinterprets the interval as describing recycling rates. Choice D inappropriately generalizes to other towns. Choice E confuses the confidence interval with a prediction interval for the same sample. Key insight: confidence levels describe the long-run success rate of the interval construction procedure across many independent samples.

Question 11

1. There is a 95% chance that the true proportion $p$ is between 0.545 and 0.635.
2. We are 95% confident that the interval from 0.545 to 0.635 captures the true proportion $p$ of all students at the school who support starting classes later. (correct answer)
3. About 95% of all students at the school support starting classes later.
4. If many random samples of 400 students were taken, 95% of the sample proportions would fall between 0.545 and 0.635.
5. If the sampling were repeated many times, 95% of the time the true proportion $p$ would change to fall between 0.545 and 0.635.

Explanation: This question tests understanding of confidence interval interpretation for a population proportion. The correct interpretation (B) states that we are 95% confident the interval captures the true proportion p. Choice A incorrectly treats the confidence level as a probability about the parameter p itself - once computed, p is fixed and either is or isn't in the interval. Choice C misinterprets the interval as describing the percentage of students who support later start times. Choice D confuses the confidence interval with a prediction interval for future sample proportions. Choice E incorrectly suggests the true proportion p changes between samples. Remember: confidence intervals describe our confidence in the method, not probability about the parameter.

Question 12

1. 95% of all subscribers prefer watching on a TV.
2. We are 95% confident that the true proportion $p$ of all subscribers who prefer TV is between 0.55 and 0.63. (correct answer)
3. There is a 95% chance that the next subscriber surveyed will prefer TV.
4. The probability that $p$ lies between 0.55 and 0.63 is 0.95.
5. If we took one more random sample of 500 subscribers, the new interval would be exactly $(0.55,\ 0.63)$ 95% of the time.

Explanation: This question evaluates understanding of confidence interval interpretation for proportions. The sample showed 295/500 = 0.59 prefer TV, yielding a 95% CI of (0.55, 0.63). Choice B correctly states we are 95% confident that the true proportion p of ALL subscribers who prefer TV is between 0.55 and 0.63. Choice A wrongly claims 95% of subscribers prefer TV (confusing confidence level with proportion). Choices C and D incorrectly treat the parameter as random or probabilistic. Choice E misunderstands confidence intervals - new samples would produce different intervals, not the same one. The confidence statement reflects our uncertainty about where the fixed but unknown parameter p lies, based on our sample evidence.

Question 13

1. There is a 93% chance that the true proportion $p$ is between 0.62 and 0.72.
2. 93% of all customers would rate their experience as "excellent."
3. If many random samples of 300 customers are taken and a 93% confidence interval is computed each time, about 93% of those intervals will contain $p$. (correct answer)
4. If we repeat the survey, 93% of the time the sample proportion will be between 0.62 and 0.72.
5. We are 93% confident that between 62% and 72% of the sampled customers (not all customers) would rate their experience as "excellent."

Explanation: This question assesses understanding of confidence level interpretation. With 201/300 = 0.67 rating "excellent," the 93% CI is (0.62, 0.72). Choice C correctly explains that if many random samples were taken and a 93% CI computed each time, about 93% of those intervals would contain p. This frequentist interpretation properly describes what 93% confidence means. Choice A incorrectly treats the parameter as random. Choice B confuses confidence level with the proportion itself. Choice D wrongly suggests the interval predicts future sample proportions. Choice E misunderstands the target of inference - we're estimating the population proportion, not describing the sample. The confidence level reflects the reliability of our interval estimation procedure over repeated sampling.

Question 14

1. We are 90% confident that the true proportion $p$ of all customers who would recommend the company is between 0.64 and 0.72. (correct answer)
2. There is a 90% chance that the interval $(0.64,\ 0.72)$ contains $p$.
3. About 90% of customers would recommend the company.
4. If many random samples of 250 customers are taken, then 90% of the sample proportions will fall between 0.64 and 0.72.
5. In 90% of all random samples of 250 customers, exactly between 64% and 72% of the sampled customers will recommend the company.

Explanation: This question assesses proper interpretation of a 90% confidence interval for a population proportion. From the sample, 170/250 = 0.68 would recommend the company, and the 90% CI is (0.64, 0.72). Choice A correctly interprets this: we are 90% confident that the true proportion p of ALL customers who would recommend the company is between 0.64 and 0.72. Choice B incorrectly treats the interval as having a probability (the parameter is fixed, not random). Choices C, D, and E misinterpret what the interval describes - it's about the population parameter, not individual customers or future samples. The confidence level refers to the long-run success rate of the interval construction method, not a probability statement about this specific interval.

Question 15

1. We are 95% confident that the true proportion $p$ of all donors who prefer email contact is between 0.50 and 0.66. (correct answer)
2. 95% of all donors prefer email contact.
3. There is a 95% chance that the next donor contacted will prefer email.
4. The probability that $p$ is between 0.50 and 0.66 is 0.95.
5. Because 95% is a high confidence level, the interval $(0.50,\ 0.66)$ must be very close to the sample proportion for any sample of 150.

Explanation: This question evaluates proper confidence interval interpretation. From 87/150 = 0.58 preferring email, the 95% CI is (0.50, 0.66). Choice A correctly states we are 95% confident that the true proportion p of ALL donors who prefer email contact is between 0.50 and 0.66. Choice B misinterprets the confidence level as a proportion. Choice C incorrectly applies the interval to predict individual outcomes. Choice D wrongly treats the parameter p as having probability. Choice E misunderstands confidence intervals - higher confidence actually produces wider intervals, not narrower ones. The key is recognizing that confidence intervals estimate fixed population parameters with a stated level of confidence in the estimation procedure.

Question 16

1. There is a 95% probability that $p$ is between 0.53 and 0.61.
2. About 95% of all district households support the tax increase.
3. We are 95% confident that the true proportion $p$ of all district households that support the tax increase is between 0.53 and 0.61. (correct answer)
4. If we sample 400 households again, 95% of the time the sample proportion will be between 0.53 and 0.61.
5. In 95% of random samples of 400 households, the interval $(0.53,\ 0.61)$ will contain the sample proportion.

Explanation: This question tests understanding of confidence interval interpretation for a population proportion. The interval (0.53, 0.61) was constructed from a sample of 400 households where 228/400 = 0.57 supported the tax increase. Choice C correctly states that we are 95% confident the true proportion p of ALL district households supporting the tax is between 0.53 and 0.61. Common misconceptions include thinking the interval gives a probability about p (Choice A), describes the population directly (Choice B), or predicts future sample proportions (Choices D and E). A confidence interval captures our uncertainty about the unknown population parameter p, not about sample statistics or individual observations.

Question 17

1. If the study were repeated many times, 99% of the computed intervals would contain the true proportion $p$. (correct answer)
2. There is a 99% probability that the true proportion $p$ is between 0.49 and 0.57.
3. 99% of adults in the city got a flu shot.
4. If many samples of 600 adults are taken, 99% of the time the sample proportion will be between 0.49 and 0.57.
5. The true proportion $p$ is between 0.49 and 0.57 for 99% of cities like this one.

Explanation: This question tests understanding of what the confidence level means in confidence interval interpretation. With 318/600 = 0.53 getting flu shots, the 99% CI is (0.49, 0.57). Choice A correctly explains that if the study were repeated many times with new random samples, about 99% of the computed intervals would contain the true proportion p. This is the frequentist interpretation of confidence level. Choice B incorrectly assigns probability to the parameter p (which is fixed). Choices C, D, and E misinterpret what the interval estimates - it's about the population proportion, not percentages of adults, sample proportions, or other cities. The key insight is that confidence describes the reliability of the interval construction procedure, not this specific interval.

Question 18

1. We are 95% confident that the true proportion $p$ of all customers who would recommend the store is between 0.76 and 0.84. (correct answer)
2. There is a 95% chance that between 76% and 84% of all customers would recommend the store.
3. If we surveyed all customers, 95% of them would fall between 0.76 and 0.84 in their responses.
4. In 95% of random samples of 500 customers, the sample proportion will be exactly 0.80.
5. The probability that $p$ is not between 0.76 and 0.84 is 0.05, since we missed $p$ with probability 0.05.

Explanation: This question tests understanding of confidence interval interpretation in a business context. The correct answer (A) properly states that we are 95% confident the true proportion p of all customers who would recommend the store is between 0.76 and 0.84. Choice B incorrectly treats this as a probability statement about p. Choice C misinterprets the interval as describing individual customer responses. Choice D incorrectly focuses on the sample proportion being exactly 0.80. Choice E, while mentioning the complement of 95%, incorrectly frames this as probability about p. Remember: once computed, the interval either contains p or it doesn't - our 95% confidence is in the method's long-run performance.

Question 19

1. We are 92% confident that the interval from 0.54 to 0.60 contains the true proportion $p$ of all adults in the state who received a flu shot. (correct answer)
2. There is a 92% probability that $p$ is between 0.54 and 0.60.
3. About 92% of adults in the state received a flu shot.
4. If we repeated the sampling many times, 92% of the time the sample proportion would be between 0.54 and 0.60.
5. Since the confidence level is 92%, the interval $(0.54,\ 0.60)$ contains 92% of the population values.

Explanation: This AP Statistics question evaluates interpretation of a confidence interval for the proportion of adults who received a flu shot. Choice A is correct: we are 92% confident the interval from 0.54 to 0.60 contains the true p. Choice B distracts by using 'probability' for p in the interval, but confidence describes the method, not a chance for the fixed p. Mini-lesson: a confidence interval's level, like 92%, means that across many repeated samples and intervals, 92% would include the true population proportion. Choice C wrongly claims 92% of the population got the shot, D misstates that sample proportions fall in the interval 92% of the time, and E confuses the interval with containing population values.

Question 20

1. If we took many random samples of 600 subscribers and built a 90% confidence interval each time, about 90% of those intervals would contain the true proportion $p$. (correct answer)
2. There is a 90% chance that the true proportion $p$ is between 0.11 and 0.15.
3. 90% of subscribers plan to cancel in the next month.
4. About 90% of samples of 600 subscribers will produce an interval of exactly $(0.11,\ 0.15)$.
5. Since the interval is $(0.11,\ 0.15)$, the sample proportion must be 0.13.

Explanation: In AP Statistics, this question assesses understanding of confidence interval interpretation for a population proportion. The correct choice, A, explains that if many random samples of 600 subscribers are taken and 90% confidence intervals built each time, about 90% of those intervals would contain the true proportion p. A frequent distractor is choice B, which wrongly assigns a probability to the true p being in the specific interval, but the interval is fixed after calculation. For a mini-lesson: a confidence interval provides a range where we're confident the parameter lies, based on the long-run success rate of the interval-capturing method; here, 90% confidence means the procedure captures p in 90% of repeated applications. Choice C misinterprets the confidence level as the proportion itself, D exaggerates by saying intervals would be exactly the same, and E incorrectly assumes the point estimate from the interval without verification.