Confidence Interval for a Population Proportion
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AP Statistics › Confidence Interval for a Population Proportion
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?
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 true proportion $p$ changes from sample to sample, and 95% of the time it will be between 0.19 and 0.41.
There is a 95% probability that the interval $(0.19, 0.41)$ contains the true proportion $p$.
About 95% of all households in the town have a pet dog.
If we repeatedly sample 60 households, 95% of those samples will have exactly 30% with a pet dog.
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.
A city randomly sampled 500 registered voters and found that 275 support a proposed public transit tax. A 90% confidence interval for the true proportion $p$ of all registered voters in the city who support the tax is $(0.52,\ 0.58)$. Which interpretation is correct?
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 $p$.
About 90% of samples of 500 voters will have sample proportion between 0.52 and 0.58.
Because the confidence level is 90%, the interval $(0.52, 0.58)$ contains exactly 90% of the population values.
About 90% of registered voters in the city support the tax.
There is a 90% probability that $p$ is between 0.52 and 0.58.
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.
A random sample of 200 customers at a grocery store found that 46 used a self-checkout lane. A 96% confidence interval for the true proportion $p$ of all customers at that store who use self-checkout is $(0.18,\ 0.28)$. Which interpretation is correct?
If a new sample of 200 customers is taken, the interval $(0.18, 0.28)$ will be produced 96% of the time.
Because 96% is close to 100%, the interval must contain the true proportion $p$.
There is a 96% chance that the true proportion $p$ is between 0.18 and 0.28.
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.
About 96% of customers at that store use self-checkout.
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.
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?
The probability that $p$ is in $(0.54, 0.66)$ is 0.94.
There is a 94% chance that the true proportion $p$ is between 0.54 and 0.66.
We are 94% confident that between 54% and 66% of all passengers on that rail line purchase tickets using a mobile app.
In repeated sampling, 94% of samples of 350 passengers will have sample proportion between 0.54 and 0.66.
94% of passengers on that rail line purchase tickets using a mobile app.
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.
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?
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.
Because the confidence level is 95%, the true proportion $p$ must be between 0.39 and 0.47.
About 95% of all high school students in the district get at least 8 hours of sleep.
There is a 95% probability that the true proportion $p$ is between 0.39 and 0.47.
If we repeated the survey many times, 95% of the students in each sample would say “yes.”
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.
A technology firm randomly samples 1,500 employees worldwide and asks whether they primarily work remotely. In the sample, 690 employees report primarily working remotely. A 97% confidence interval for the true proportion $p$ of all employees worldwide who primarily work remotely is $(0.43,\ 0.49)$. Which interpretation is correct?
We are 97% confident that between 43% and 49% of the sampled employees primarily work remotely.
97% of all employees worldwide primarily work remotely.
The confidence interval means the true proportion $p$ will be between 0.43 and 0.49 for 97% of the employees.
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 $p$.
There is a 97% chance that the true proportion $p$ is between 0.43 and 0.49.
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.
A streaming service randomly sampled 500 subscribers to estimate the proportion $p$ who watched a particular new series in its first week. In the sample, 165 subscribers watched it. A 96% confidence interval for $p$ was computed as $(0.30, 0.36)$. Which interpretation is correct?
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.
There is a 96% probability that the true proportion $p$ is between 0.30 and 0.36.
If we sample again, 96% of the time the sample proportion will be exactly 0.33, the midpoint of the interval.
About 96% of subscribers watched the series in its first week.
Because the confidence interval is 0.30 to 0.36, exactly 30% to 36% of the 500 sampled subscribers watched the series.
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.
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?
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.
There is a 95% chance that the true proportion $p$ equals 0.61.
If the sampling were repeated, 95% of the time the sample proportion would be between 0.58 and 0.64 exactly.
95% of all adults in the state approve of the governor.
Because the confidence interval is narrow, it must contain the true proportion $p$.
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.
A university randomly sampled 350 undergraduate students to estimate the proportion $p$ who have taken at least one online course. In the sample, 210 students had taken an online course. A 92% confidence interval for $p$ was reported as $(0.55, 0.65)$. Which interpretation is correct?
We can be 92% confident that 92% of students are between 0.55 and 0.65 likely to have taken an online course.
If the sampling method is repeated many times, about 92% of the confidence intervals constructed this way will contain the true proportion $p$.
There is a 92% probability that the sample proportion is between 0.55 and 0.65.
About 92% of all undergraduates have taken an online course.
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.
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?
There is a 92% chance that $p$ is between 0.54 and 0.70.
About 92% of households in the town regularly recycle.
If the same 150 households were surveyed again, 92% of the time the sample proportion would be between 0.54 and 0.70.
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.
In 92% of all towns, the proportion that regularly recycle is 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.