Confidence Interval for a Population Proportion - AP Statistics

1. A state agency randomly sampled 1,000 drivers to estimate the proportion $p$ who support a new hands-free law. In the sample, 610 drivers supported the law. A 95% confidence interval for $p$ was reported as $(0.58, 0.64)$. Which interpretation is correct?
   Answer: If the agency repeatedly took random samples of 1,000 drivers and constructed 95% confidence intervals, about 95% of those intervals would contain the true proportion $p$. This question tests confidence interval interpretation for policy support. The sample proportion is 610/1000 = 0.61, with a 95% CI of (0.58, 0.64). The correct answer (C) properly states that if the agency repeatedly sampled and constructed 95% CIs, about 95% would contain the true proportion. Choice A incorrectly refers to the sample - we know exactly 61% of the sample supported the law. Choice B incorrectly assigns probability to the parameter. Choice D confuses the confidence level with the support rate. Choice E misinterprets the interval as providing certainty about individual probabilities. Confidence intervals quantify uncertainty about population parameters through the reliability of the construction method.
2. A random sample of 60 households in a town found that 18 have a pet dog. A 95% confidence interval for the true proportion $p$ of all households in the town that have a pet dog is $(0.19,\ 0.41)$. Which interpretation is correct?
   Answer: 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. 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.
3. A streaming service takes a simple random sample of 600 subscribers and asks whether they plan to cancel in the next month. In the sample, 78 say they plan to cancel. A 90% confidence interval for the true proportion $p$ of all subscribers who plan to cancel in the next month is $(0.11,\ 0.15)$. Which interpretation is correct?
   Answer: 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 $p$. 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.
4. A random sample of 120 smartphones from a shipment found that 9 have a defective screen. A 90% confidence interval for the true proportion $p$ of all smartphones in the shipment with defective screens is $(0.03,\ 0.12)$. Which interpretation is correct?
   Answer: If many random samples of 120 smartphones are taken and a 90% confidence interval is computed each time, about 90% of those intervals will contain the true proportion $p$. This AP Statistics item tests interpreting a confidence interval for a population proportion. The correct choice explains that if many samples of 120 smartphones are taken and 90% intervals computed, about 90% will contain the true p, highlighting the long-run method performance. Distractor D suggests 90% of samples will have defective rates between 3% and 12%, which is incorrect as it applies this specific interval to future sample proportions, not accounting for interval variability. Mini-lesson: CIs mean the construction process captures the parameter in 90% of repetitions, providing a range with associated confidence. This clarifies the distinction between probability for the method and the fixed parameter. Understanding this is crucial for valid statistical conclusions.
5. A restaurant chain randomly samples 300 customers and asks whether their order was accurate. In the sample, 279 customers report an accurate order. A 90% confidence interval for the true proportion $p$ of all customers whose orders are accurate is $(0.90,\ 0.96)$. Which interpretation is correct?
   Answer: We are 90% confident that between 0.90 and 0.96 of all customers at the chain receive accurate orders. This AP Statistics question involves interpreting a confidence interval for order accuracy proportion. Choice A is right: we are 90% confident that between 0.90 and 0.96 of all customers receive accurate orders. Choice B is a distractor, incorrectly stating a probability for p in the interval. Mini-lesson: confidence intervals estimate the population proportion with a level reflecting the method's reliability over many samples. Choice C applies 90% to the sample, D claims identical intervals in repetitions, and E suggests higher likelihood at the center, which isn't part of standard interpretation.
6. A nonprofit randomly sampled 150 donors and found that 39 prefer to donate monthly rather than yearly. A 92% confidence interval for the true proportion $p$ of all donors who prefer monthly giving is $(0.20,\ 0.33)$. Which interpretation is correct?
   Answer: We are 92% confident that between 20% and 33% of all donors prefer monthly giving. This question examines understanding of a 92% confidence interval for a population proportion. The correct answer (B) appropriately states we are 92% confident that between 20% and 33% of all donors prefer monthly giving. Choice A nonsensically relates confidence to computational accuracy. Choice C incorrectly suggests the true proportion p changes across samples. Choice D makes an absurd claim about sample proportions equaling p. Choice E wrongly treats the confidence level as a probability about the parameter p. Key principle: confidence intervals express our confidence in the interval construction method, not probabilities about fixed but unknown parameters.
7. A company randomly sampled 1,200 employees and found that 312 report working remotely at least 3 days per week. A 90% confidence interval for the true proportion $p$ of all employees at the company who work remotely at least 3 days per week is $(0.24,\ 0.28)$. Which interpretation is correct?
   Answer: We are 90% confident that the interval $(0.24,\ 0.28)$ contains the true proportion $p$ of employees who work remotely at least 3 days per week. This question tests interpretation of a 90% confidence interval for a population proportion. The correct answer (A) appropriately states we are 90% confident the interval contains the true proportion p. Choice B incorrectly treats the confidence level as a probability about the parameter p. Choice C wrongly suggests p varies and will be captured in this interval in 90% of samples. Choice D incorrectly claims the true proportion p changes across samples. Choice E misinterprets the interval as describing 90% of employees. Remember: confidence intervals express our confidence in the method that produced the interval, not probabilities about parameters or predictions about future samples.
8. A school district surveyed a simple random sample of 400 high school students about whether they get at least 8 hours of sleep on a typical school night. In the sample, 172 students said "yes." A 95% confidence interval for the true proportion $p$ of all high school students in the district who get at least 8 hours of sleep is $(0.39,\ 0.47)$. Which interpretation is correct?
   Answer: 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. 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.
9. A university randomly sampled 250 undergraduate students and found that 160 said they prefer taking exams on a computer rather than on paper. A 99% confidence interval for the true proportion $p$ of all undergraduates at the university who prefer computer-based exams is $(0.55,\ 0.73)$. Which interpretation is correct?
   Answer: We are 99% confident that the interval $(0.55,\ 0.73)$ contains the true proportion $p$ of all undergraduates who prefer computer-based exams. This question evaluates understanding of a 99% confidence interval interpretation. The correct answer (C) appropriately expresses confidence that the interval contains the true proportion p. Choice A incorrectly links the confidence level to the actual percentage preferring computer exams. Choice B wrongly treats the confidence level as a probability about the parameter p. Choice D mistakenly suggests the true proportion p will fall in this interval in future samples. Choice E incorrectly claims the sample proportion equals p and confuses confidence with probability. Remember: confidence intervals express certainty about the method used to construct the interval, based on repeated sampling theory.
10. A city randomly sampled 250 registered voters to estimate the proportion $p$ who support a proposed public transit tax. In the sample, 120 voters supported the tax. A 90% confidence interval for $p$ was reported as $(0.43, 0.53)$. Which interpretation is correct?
    Answer: We are 90% confident that the interval from 0.43 to 0.53 captures the true proportion $p$ of all registered voters in the city who support the tax. This question assesses proper interpretation of a confidence interval for a proportion. The sample proportion is 120/250 = 0.48, and the 90% CI is (0.43, 0.53). The correct answer (A) properly states that we are 90% confident the interval captures the true proportion of all registered voters who support the tax. Choice B incorrectly applies the confidence level to individual voters rather than the interval's coverage probability. Choice C misunderstands repeated sampling - new intervals would vary, not all equal (0.43, 0.53). The confidence level refers to the long-run proportion of intervals that would contain the true parameter if we repeated the sampling process, not to any probability about individual voters or specific outcomes.
11. A nonprofit wants to estimate the proportion $p$ of residents in a region who have volunteered at least once in the past year. In a random sample of 360 residents, 99 report volunteering. A 95% confidence interval for $p$ is $(0.23, 0.32)$. Which interpretation is correct?
    Answer: If many random samples of 360 residents were selected and a 95% confidence interval computed each time, about 95% of those intervals would contain the true proportion $p$ of residents in the region who volunteered at least once in the past year. This question tests understanding of the repeated sampling interpretation of confidence intervals. The correct answer (A) accurately describes that if many random samples were taken and confidence intervals computed, about 95% of those intervals would contain the true proportion p. Choice B incorrectly assigns probability to the parameter p. Choice C misinterprets the confidence level as describing volunteer percentages. Choice D incorrectly suggests p changes between samples. Choice E, while mentioning repeated sampling, confuses the confidence interval with a prediction about sample proportions. The fundamental concept: confidence levels describe the long-run success rate of the interval construction procedure, not predictions about future samples or probability about p.
12. A health organization takes a simple random sample of 800 adults in a state and asks whether they received a flu shot this season. In the sample, 456 adults report receiving a flu shot. A 92% confidence interval for the true proportion $p$ of all adults in the state who received a flu shot is $(0.54,\ 0.60)$. Which interpretation is correct?
    Answer: 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. 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.
13. A school district randomly sampled 240 high school students and found that 156 reported getting at least 8 hours of sleep on a typical school night. A 95% confidence interval for the true proportion $p$ of all high school students in the district who get at least 8 hours of sleep is $(0.59,\ 0.71)$. Which interpretation is correct?
    Answer: We are 95% confident that the interval from 0.59 to 0.71 captures the true proportion $p$ of all high school students in the district who get at least 8 hours of sleep. 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 0.59 to 0.71 captures the true proportion p of all high school students in the district who get at least 8 hours of sleep, meaning the method used to create such intervals will contain the true p in 95% of repeated samples. A common distractor is choice A, which incorrectly states there is a 95% chance that p is between 0.59 and 0.71, but this is wrong because p is a fixed value, and the probability applies to the interval's construction, not to p itself. In a mini-lesson on confidence intervals, remember that a 95% CI does not mean the true parameter has a 95% probability of being in this specific interval; instead, it reflects the long-run proportion of times the interval would capture the parameter if the sampling were repeated infinitely. This distinction helps avoid confusing the fixed nature of p with the variability of the sample-based interval. Understanding this ensures accurate communication of statistical confidence without implying probability about the parameter.
14. A school newspaper wants to estimate the proportion $p$ of all students at the school who support starting classes later. In a random sample of 400 students, 236 said they support a later start time. A 95% confidence interval for $p$ is $(0.545, 0.635)$. Which interpretation is correct?
    Answer: 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. 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.
15. A news organization takes a simple random sample of 1,200 adults nationwide and asks whether they approve of a public official's job performance. In the sample, 624 approve. A 95% confidence interval for the true proportion $p$ of all adults nationwide who approve is $(0.49,\ 0.55)$. Which interpretation is correct?
    Answer: We are 95% confident that the true proportion $p$ of all adults nationwide who approve is between 0.49 and 0.55. The skill in this AP Statistics question is proper interpretation of a confidence interval for approval proportion. Choice B correctly states we are 95% confident the true p is between 0.49 and 0.55. Choice D distracts by using 'probability' directly for p in the interval, which is incorrect for a fixed parameter. Mini-lesson: a 95% confidence level means the interval construction method succeeds in including the true proportion in 95% of repeated samples. Choice A misapplies to the sample proportion, C equates 95% to the population, and E is unrelated to interpretation (it's about interval width).
16. An environmental group estimates the proportion $p$ of households in a town that regularly recycle. In a random sample of 150 households, 93 say they regularly recycle. A 92% confidence interval for $p$ is $(0.54, 0.70)$. Which interpretation is correct?
    Answer: 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. 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.
17. A city council randomly sampled 350 registered voters to estimate the proportion $p$ of all registered voters in the city who favor building a new park. In the sample, 182 favored the park. A 92% confidence interval for $p$ is $(0.47,\ 0.57)$. Which interpretation is correct?
    Answer: If many random samples of 350 voters are taken and a 92% confidence interval is built each time, about 92% of those intervals will contain $p$. This question tests the frequentist interpretation of confidence level. With 182/350 = 0.52 favoring the park, the 92% CI is (0.47, 0.57). Choice A correctly explains that if many random samples were taken and a 92% CI computed each time, about 92% of those intervals would contain the true proportion p. This captures what "92% confidence" means - it's about the long-run performance of the interval construction method. Choice B incorrectly assigns probability to the parameter. Choices C and D confuse the confidence level with proportions. Choice E misinterprets the interval as predicting future sample proportions rather than estimating the population parameter. Understanding this frequentist interpretation is crucial for proper statistical inference.
18. A random sample of 350 passengers on a commuter rail line found that 210 purchased their ticket using a mobile app. A 94% confidence interval for the true proportion $p$ of all passengers on that rail line who purchase tickets using a mobile app is $(0.54,\ 0.66)$. Which interpretation is correct?
    Answer: We are 94% confident that between 54% and 66% of all passengers on that rail line purchase tickets using a mobile app. 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.
19. A random sample of 1,000 adults in a state found that 610 approve of the governor's job performance. A 95% confidence interval for the true proportion $p$ of all adults in the state who approve is $(0.58,\ 0.64)$. Which interpretation is correct?
    Answer: 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. 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.
20. A streaming service surveys a random sample of 600 subscribers to estimate the proportion $p$ of all subscribers who primarily watch on a phone. The company reports a 99% confidence interval of $(0.18, 0.26)$ for $p$. Which interpretation is correct?
    Answer: If many random samples of 600 subscribers were taken and a 99% confidence interval computed each time, about 99% of those intervals would contain the true population proportion $p$. This question evaluates understanding of what a 99% confidence level means. The correct answer (B) properly describes the long-run behavior of the confidence interval method - if we repeated the sampling process many times, about 99% of the resulting intervals would contain the true proportion p. Choice A incorrectly refers to the sample rather than the population. Choice C misinterprets the interval as a prediction about individual subscribers. Choice D makes no logical sense connecting the interval bounds to usage percentages. Choice E incorrectly treats confidence as probability about the parameter p. Remember: confidence levels describe the success rate of the interval construction method over many repetitions.

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