Confidence Interval for a Population Mean
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AP Statistics › Confidence Interval for a Population Mean
A researcher selected a simple random sample of 18 adult housecats from a city and measured their weights (in pounds). A 95% confidence interval for the population mean cat weight was $(9.6, 11.2)$ pounds. Which interpretation is correct?
There is a 95% chance that the true mean weight of adult housecats in the city is between 9.6 and 11.2 pounds.
In 95% of all random samples of 18 cats, at least 95% of the cat weights will fall between 9.6 and 11.2 pounds.
About 95% of adult housecats in the city weigh between 9.6 and 11.2 pounds.
Because the sample size is 18, the confidence interval guarantees the mean is between 9.6 and 11.2 pounds.
We are 95% confident that the interval from 9.6 to 11.2 pounds captures the true population mean weight of adult housecats in the city.
Explanation
This question focuses on interpreting a confidence interval for a population mean in AP Statistics. The correct interpretation is choice B, which says we are 95% confident that the interval from 9.6 to 11.2 pounds captures the true population mean weight of adult housecats. Choice C is a distractor that incorrectly assigns a 95% probability to the mean being in the interval, treating the unknown but fixed mean as a random variable. For a mini-lesson on mean confidence intervals, they are built around the sample mean with a margin of error accounting for sample size and variability, using t-critical values for small samples. The confidence level indicates how often the interval construction method succeeds in including the true mean in repeated sampling. This helps clarify that the interval is about the mean, not individual values or a guarantee.
A random sample of 40 students at a large university recorded the number of minutes they spent studying the night before a midterm. A 95% confidence interval for the population mean study time was $(62, 78)$ minutes. Which interpretation is correct?
About 95% of all students at the university studied between 62 and 78 minutes the night before the midterm.
If many random samples of 40 students were taken and a 95% confidence interval computed each time, about 95% of those intervals would contain the true population mean study time.
The true population mean study time is guaranteed to be between 62 and 78 minutes because the confidence level is 95%.
In 95% of random samples of 40 students, the sample mean study time will fall between 62 and 78 minutes.
There is a 95% probability that the population mean study time is between 62 and 78 minutes.
Explanation
This question assesses the skill of interpreting a confidence interval for a population mean in AP Statistics. The correct interpretation, choice C, states that if many random samples are taken and 95% confidence intervals are computed each time, about 95% of those intervals would contain the true population mean study time. A common distractor is choice A, which incorrectly assigns a probability to the population mean being in the interval, treating the fixed mean as random after the interval is calculated. In a mini-lesson on confidence intervals for means, remember that a 95% confidence interval means that the method used to construct the interval will capture the true population mean in 95% of repeated samples over the long run. It does not mean there's a 95% chance the mean is in this specific interval or that 95% of data points fall within it. This distinction helps avoid misinterpreting the interval as a prediction interval or a probabilistic statement about the parameter itself.
A random sample of 12 different days was selected, and the amount of rainfall (in inches) was recorded for each day in a region. A 90% confidence interval for the population mean daily rainfall was $(0.18, 0.34)$ inches. Which interpretation is correct?
We are 90% confident that the true population mean daily rainfall in the region is between 0.18 and 0.34 inches.
The probability that the true population mean daily rainfall is between 0.18 and 0.34 inches is 0.90.
In repeated sampling, 90% of the time the population mean will change to fall within $(0.18, 0.34)$.
There is a 90% chance that exactly 90% of days have rainfall between 0.18 and 0.34 inches.
Because 12 days were sampled, the interval $(0.18, 0.34)$ contains the mean rainfall for those 12 days.
Explanation
This question tests the skill of interpreting a confidence interval for a population mean in AP Statistics. The correct choice, A, states that we are 90% confident the true population mean daily rainfall is between 0.18 and 0.34 inches, appropriately conveying the interval's reliability. Choice C is a distractor that incorrectly assigns a 0.90 probability to the mean being in the interval, which doesn't align with the fixed parameter. For a mini-lesson on mean confidence intervals, especially with small samples like 12, use the t-distribution to account for added uncertainty in the standard deviation estimate. The confidence level reflects the proportion of such intervals that would contain the true mean in repeated sampling. This phrasing helps distinguish it from predictions about daily rainfall amounts.
An environmental scientist took a random sample of 45 water samples from a lake and measured nitrate concentration (in mg/L). A 95% confidence interval for the population mean nitrate concentration was $(3.1, 3.8)$ mg/L. Which interpretation is correct?
If the scientist repeatedly took random samples of 45 water samples and constructed 95% confidence intervals, about 95% of those intervals would contain the true population mean nitrate concentration.
The interval $(3.1, 3.8)$ means the nitrate concentration is between 3.1 and 3.8 mg/L for 95% of the lake.
There is a 95% chance that the true population mean nitrate concentration is between 3.1 and 3.8 mg/L.
In 95% of samples, the sample mean nitrate concentration will be between 3.1 and 3.8 mg/L.
95% of all water samples from the lake have nitrate concentrations between 3.1 and 3.8 mg/L.
Explanation
This question assesses the skill of confidence interval interpretation for a population mean in AP Statistics. Choice C is correct, stating that if random samples of 45 water samples are repeatedly taken and 95% intervals constructed, about 95% would contain the true population mean nitrate concentration. A distractor like choice B incorrectly suggests a 95% chance for the mean in this interval. In a mini-lesson on mean confidence intervals, the interval is the sample mean ± margin of error, with the level indicating success in repeated applications. This repeated sampling view clarifies it's about the method, not individual samples or probabilities on the mean. It prevents confusion with distributions of the data itself.
A restaurant owner wants to estimate the mean amount (in dollars) that customers spend per visit. A random sample of 80 customer receipts from the past month was selected, and a 90% confidence interval for the population mean spending was computed as $(12.40, 14.10)$. Which interpretation is correct?
About 90% of all customer receipts are between $12.40 and $14.10.
We are 90% confident that the true mean amount customers spend per visit is between $12.40 and $14.10.
There is a 90% probability that the population mean spending is between $12.40 and $14.10.
If another sample of 80 receipts is taken, the sample mean will fall between $12.40 and $14.10 with probability 0.90.
About 10% of all 90% confidence intervals computed from samples of 80 receipts will contain the true mean spending.
Explanation
This question tests understanding of a 90% CI for mean customer spending. The correct answer (B) states we are 90% confident the true mean is between $12.40 and $14.10. Choice A incorrectly assigns probability to the population parameter. Choice C confuses the interval for the mean with individual receipts. Choice D misinterprets the interval as predicting future sample means. Choice E reverses the interpretation - 90% of intervals contain the true mean, not 10%. Remember that once computed, a confidence interval either contains the true parameter or it doesn't - there's no probability involved.
A school district wants to estimate the mean time (in minutes) it takes high school students to complete a certain standardized math assessment. A random sample of 30 students took the assessment, and a 98% confidence interval for the population mean completion time was found to be $(47.5,\ 52.9)$ minutes. Which interpretation is correct?
About 98% of all possible sample means from samples of 30 students are between 47.5 and 52.9 minutes.
We are 98% confident that the true mean completion time for all students in the district is between 47.5 and 52.9 minutes.
If the district repeated the sampling process, 98% of the time the interval $(47.5,\ 52.9)$ would be produced again.
There is a 98% chance that the true mean completion time is between 47.5 and 52.9 minutes because the mean is random.
There is a 98% probability that any randomly selected student will complete the assessment between 47.5 and 52.9 minutes.
Explanation
This question asks about interpreting a 98% CI for mean completion time. The correct answer (A) properly states we are 98% confident the true mean is between 47.5 and 52.9 minutes. Choice B incorrectly applies the interval to individual students. Choice C misunderstands what repeating the process means - we'd get different intervals, not the same one. Choice D wrongly suggests the population mean is random. Choice E confuses the interval with the sampling distribution of the sample mean. Confidence intervals estimate fixed population parameters, not predict individual values or sample statistics.
A consumer group wants to estimate the mean lifetime (in months) of a certain brand of rechargeable battery. A random sample of 15 batteries was tested to failure, and a 95% confidence interval for the population mean lifetime was computed as $(18.2,\ 24.6)$ months. Which interpretation is correct?
The probability that a randomly selected battery lasts longer than 24.6 months is 0.05.
If the test were repeated many times with samples of 15 batteries, about 95% of the confidence intervals constructed would contain the true mean battery lifetime.
There is a 95% chance that the true mean battery lifetime is between 18.2 and 24.6 months.
About 95% of batteries of this brand last between 18.2 and 24.6 months.
Because the confidence interval is 95%, the true mean lifetime will be between 18.2 and 24.6 months for 95% of future years.
Explanation
This problem involves interpreting a 95% CI for mean battery lifetime. The correct answer (B) states that if we repeated the sampling process many times, about 95% of the resulting confidence intervals would contain the true mean. Choice A incorrectly applies the interval to individual batteries. Choice C wrongly assigns probability to the fixed parameter. Choice D makes an unrelated claim about individual batteries. Choice E nonsensically suggests the population parameter changes over time. The key insight is that confidence refers to the procedure's long-run success rate, not probability about a specific interval.
A public health official wants to estimate the mean number of servings of vegetables eaten per day by adults in a county. A random sample of 100 adults was surveyed, and a 97% confidence interval for the population mean was reported as $(2.1,\ 2.8)$ servings. Which interpretation is correct?
About 97% of adults in the county eat between 2.1 and 2.8 servings of vegetables per day.
We are 97% confident that the true mean number of servings of vegetables eaten per day by adults in the county is between 2.1 and 2.8.
If 97% confidence intervals are repeatedly constructed, 97% of the time the population mean will change to be inside the interval.
Because the confidence level is 97%, exactly 97 of the 100 sampled adults must have reported between 2.1 and 2.8 servings per day.
There is a 97% chance that the sample mean is between 2.1 and 2.8 servings.
Explanation
This problem asks about interpreting a 97% CI for mean vegetable servings. The correct answer (B) properly states we are 97% confident the true mean is between 2.1 and 2.8 servings. Choice A incorrectly applies the interval to individual adults. Choice C wrongly suggests the sample mean is uncertain after calculation. Choice D nonsensically claims the population parameter changes. Choice E makes an incorrect claim about the sample data. A confidence interval provides a range estimate for the population mean based on sample data, with the confidence level indicating the procedure's reliability.
A coffee shop manager wants to estimate the mean amount of time customers wait in line during the morning rush. A random sample of 25 mornings was selected, and the manager recorded the average wait time (in minutes) for customers on each sampled morning. A 90% confidence interval for the population mean wait time was calculated as $(3.2,\ 4.5)$ minutes. Which interpretation is correct?
We are 90% confident that the true mean wait time during the morning rush is between 3.2 and 4.5 minutes.
About 90% of individual customer wait times are between 3.2 and 4.5 minutes.
If another random sample of 25 mornings is taken, the new confidence interval will definitely be between 3.2 and 4.5 minutes.
About 10% of all possible 90% confidence intervals will contain the true mean wait time.
There is a 90% probability that the mean wait time during the morning rush is between 3.2 and 4.5 minutes.
Explanation
This question asks about interpreting a 90% confidence interval for mean wait time. The correct answer (A) properly states that we are 90% confident the true mean wait time is between 3.2 and 4.5 minutes. Choice B incorrectly assigns probability to the population parameter after the interval is computed. Choice C confuses the interval for the mean with individual wait times. Choice D makes an incorrect claim about future samples. Choice E reverses the interpretation - 90% of intervals contain the true mean, not 10%. A confidence interval provides a range of plausible values for the population parameter based on our sample data.
A university wellness center wants to estimate the mean number of hours of sleep per night for all first-year students at the university. A random sample of 40 first-year students reported their sleep from the previous night, and a one-sample $t$ interval was constructed. The resulting 95% confidence interval for the population mean sleep time was $(6.6,\ 7.4)$ hours. Which interpretation is correct?
If many random samples of 40 first-year students were taken and a 95% confidence interval were computed each time, about 95% of those intervals would contain the true population mean sleep time.
There is a 95% chance that the true population mean sleep time is between 6.6 and 7.4 hours.
If the study were repeated, 95% of the sample means would fall between 6.6 and 7.4 hours.
About 95% of all first-year students sleep between 6.6 and 7.4 hours per night.
The population mean sleep time for first-year students is 7.0 hours with probability 0.95.
Explanation
This question tests understanding of confidence interval interpretation for a population mean. The correct interpretation (B) states that if we repeated the sampling process many times and computed a 95% CI each time, about 95% of those intervals would contain the true population mean. Choice A incorrectly treats the population parameter as random - once computed, the interval either contains the true mean or it doesn't. Choice C confuses the confidence interval for the mean with a prediction interval for individual values. Choice D incorrectly assigns probability to a fixed parameter. Choice E misinterprets the interval as capturing future sample means rather than the population mean.