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?
About 95% of all households in the town have a pet dog.
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.
If we repeatedly sample 60 households, 95% of those samples will have exactly 30% with a pet dog.
There is a 95% probability that the interval $(0.19, 0.41)$ contains the true proportion $p$.
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.
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?
There is a 90% probability that $p$ is between 0.52 and 0.58.
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.
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 registered voters in the city support the tax.
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?
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.
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 $p$.
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?
94% of 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.
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.
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.
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?
Because the confidence level is 95%, the true proportion $p$ must be between 0.39 and 0.47.
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.
If we repeated the survey many times, 95% of the students in each sample would say “yes.”
There is a 95% probability that the true proportion $p$ is between 0.39 and 0.47.
About 95% of all high school students in the district get at least 8 hours of sleep.
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.
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.
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.
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?
If we sample again, 96% of the time the sample proportion will be exactly 0.33, the midpoint of the interval.
Because the confidence interval is 0.30 to 0.36, exactly 30% to 36% of the 500 sampled subscribers watched the series.
About 96% of subscribers watched the series in its first week.
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.
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?
Because the confidence interval is narrow, it must contain the true proportion $p$.
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.
If the sampling were repeated, 95% of the time the sample proportion would be between 0.58 and 0.64 exactly.
There is a 95% chance that the true proportion $p$ equals 0.61.
95% of all adults in the state approve of the governor.
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?
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.
We can be 92% confident that 92% of students are between 0.55 and 0.65 likely to 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?
About 92% of households in the town regularly recycle.
There is a 92% chance that $p$ is 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.
If the same 150 households were surveyed again, 92% of the time the sample proportion would be between 0.54 and 0.70.
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.