AP Statistics Quiz: Confidence Interval For A Population Mean

What this quiz covers

This quiz focuses on Confidence Interval For A Population Mean, 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 adult housecats in the city weigh between 9.6 and 11.2 pounds.
2. 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. (correct answer)
3. There is a 95% chance that the true mean weight of adult housecats in the city is between 9.6 and 11.2 pounds.
4. 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.
5. Because the sample size is 18, the confidence interval guarantees the mean is between 9.6 and 11.2 pounds.

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.

Question 2

1. There is a 95% probability that the population mean study time is between 62 and 78 minutes.
2. About 95% of all students at the university studied between 62 and 78 minutes the night before the midterm.
3. 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. (correct answer)
4. In 95% of random samples of 40 students, the sample mean study time will fall between 62 and 78 minutes.
5. The true population mean study time is guaranteed to be between 62 and 78 minutes because the confidence level is 95%.

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.

Question 3

1. We are 90% confident that the true population mean daily rainfall in the region is between 0.18 and 0.34 inches. (correct answer)
2. There is a 90% chance that exactly 90% of days have rainfall between 0.18 and 0.34 inches.
3. The probability that the true population mean daily rainfall is between 0.18 and 0.34 inches is 0.90.
4. In repeated sampling, 90% of the time the population mean will change to fall within $(0.18, 0.34)$.
5. 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.

Question 4

1. 95% of all water samples from the lake have nitrate concentrations between 3.1 and 3.8 mg/L.
2. There is a 95% chance that the true population mean nitrate concentration is between 3.1 and 3.8 mg/L.
3. 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. (correct answer)
4. In 95% of samples, the sample mean nitrate concentration will be between 3.1 and 3.8 mg/L.
5. The interval $(3.1, 3.8)$ means the nitrate concentration is between 3.1 and 3.8 mg/L for 95% of the lake.

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.

Question 5

1. There is a 90% probability that the population mean spending is between $12.40 and $14.10.
2. We are 90% confident that the true mean amount customers spend per visit is between $12.40 and $14.10. (correct answer)
3. About 90% of all customer receipts are between $12.40 and $14.10.
4. If another sample of 80 receipts is taken, the sample mean will fall between $12.40 and $14.10 with probability 0.90.
5. 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.

Question 6

1. We are 98% confident that the true mean completion time for all students in the district is between 47.5 and 52.9 minutes. (correct answer)
2. There is a 98% probability that any randomly selected student will complete the assessment between 47.5 and 52.9 minutes.
3. If the district repeated the sampling process, 98% of the time the interval $(47.5,\ 52.9)$ would be produced again.
4. There is a 98% chance that the true mean completion time is between 47.5 and 52.9 minutes because the mean is random.
5. About 98% of all possible sample means from samples of 30 students are 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.

Question 7

1. About 95% of batteries of this brand last between 18.2 and 24.6 months.
2. 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. (correct answer)
3. There is a 95% chance that the true mean battery lifetime is between 18.2 and 24.6 months.
4. The probability that a randomly selected battery lasts longer than 24.6 months is 0.05.
5. 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.

Question 8

1. About 97% of adults in the county eat between 2.1 and 2.8 servings of vegetables per day.
2. 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. (correct answer)
3. There is a 97% chance that the sample mean is between 2.1 and 2.8 servings.
4. If 97% confidence intervals are repeatedly constructed, 97% of the time the population mean will change to be inside the interval.
5. 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.

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.

Question 9

1. We are 90% confident that the true mean wait time during the morning rush is between 3.2 and 4.5 minutes. (correct answer)
2. There is a 90% probability that the mean wait time during the morning rush is between 3.2 and 4.5 minutes.
3. About 90% of individual customer wait times are between 3.2 and 4.5 minutes.
4. If another random sample of 25 mornings is taken, the new confidence interval will definitely be between 3.2 and 4.5 minutes.
5. About 10% of all possible 90% confidence intervals will contain the true mean wait time.

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.

Question 10

1. There is a 95% chance that the true population mean sleep time is between 6.6 and 7.4 hours.
2. 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. (correct answer)
3. About 95% of all first-year students sleep between 6.6 and 7.4 hours per night.
4. The population mean sleep time for first-year students is 7.0 hours with probability 0.95.
5. If the study were repeated, 95% of the sample means would fall between 6.6 and 7.4 hours.

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.

Question 11

1. There is a 99% probability that the next user's load time will be between 2.10 and 2.55 seconds.
2. If 100 different random samples of 60 load times were taken, about 99 of the resulting intervals would contain the sample mean from their own sample.
3. We are 99% confident that the true population mean load time for all users is between 2.10 and 2.55 seconds. (correct answer)
4. About 99% of all users have load times between 2.10 and 2.55 seconds.
5. The true population mean load time is between 2.10 and 2.55 seconds in 99% of all possible populations.

Explanation: This question evaluates the skill of correctly interpreting a confidence interval for a population mean in AP Statistics. Choice C is correct, stating that we are 99% confident the true population mean load time is between 2.10 and 2.55 seconds, which is a standard way to express the interval's meaning without implying probability on the fixed mean. A common distractor is choice D, which misapplies the interval to individual users rather than the mean, confusing it with a prediction interval. In a mini-lesson on confidence intervals for means, these intervals estimate the population mean with a specified level of confidence based on sample data and variability. The 'we are X% confident' phrasing captures the reliability of the method, meaning that if we repeat the process, X% of intervals will include the true mean. Always distinguish this from probabilities about future observations or the parameter itself to interpret accurately.

Question 12

1. There is a 98% probability that the true population mean calories per serving is between 112 and 118.
2. We are 98% confident that the true population mean calories per serving for this cereal brand is between 112 and 118. (correct answer)
3. About 98% of cereal servings have between 112 and 118 calories.
4. About 98% of random samples of size 30 will produce a sample mean between 112 and 118 calories.
5. The confidence interval shows that the mean calories per serving is exactly 115 calories with 98% confidence.

Explanation: This question focuses on interpreting a confidence interval for a population mean in AP Statistics. The correct interpretation, choice B, is that we are 98% confident the true population mean calories per serving is between 112 and 118. Choice A distracts by assigning a 98% probability to the mean being in the interval, which is incorrect for the fixed parameter. For a mini-lesson on confidence intervals for means, they estimate the population average with a range based on sample data and variability. The 'confident' language encapsulates the method's reliability without probabilistic claims on the parameter. Distinguishing this from intervals for individual servings is crucial.

Question 13

1. We are 90% confident that the interval from 118 to 134 customers contains the true population mean number of customers per day. (correct answer)
2. There is a 90% chance that the population mean number of customers per day is between 118 and 134.
3. If we repeated the sampling many times, 90% of the time the population mean would change to fall within (118, 134).
4. About 90% of all days have between 118 and 134 customers.
5. About 90% of sample means from samples of 18 days will fall between 118 and 134.

Explanation: This question assesses proper interpretation of a confidence interval for a mean. The correct answer (A) properly states that we are 90% confident the interval contains the true population mean number of customers per day. Choice B incorrectly assigns probability to the parameter after the interval is computed. Choice C wrongly suggests the population mean changes with repeated sampling. Choice D misinterprets the interval as describing individual daily values rather than the mean. Choice E incorrectly describes where sample means would fall. Key concept: confidence intervals estimate population parameters, not individual observations or sample statistics.

Question 14

1. There is a 92% chance that the true population mean number of push-ups is between 21.4 and 26.8.
2. We are 92% confident that the true mean number of push-ups adults in the city can do in one minute is between 21.4 and 26.8. (correct answer)
3. About 92% of adults in the city can do between 21.4 and 26.8 push-ups in one minute.
4. If many 92% confidence intervals are computed from repeated samples of 50 adults, about 8% of those intervals will contain the true population mean.
5. Because the confidence level is 92%, the sample mean must be between 21.4 and 26.8 push-ups with probability 0.92.

Explanation: This problem tests understanding of a 92% confidence interval for mean push-ups. The correct answer (B) states we are 92% confident the true mean is between 21.4 and 26.8 push-ups. Choice A incorrectly treats the population parameter as random. Choice C confuses the interval for the mean with individual performance. Choice D reverses the interpretation - 92% of intervals contain the true mean, not 8%. Choice E misunderstands what the confidence level refers to. The confidence level describes the long-run success rate of the interval estimation procedure, not probability statements about specific intervals.

Question 15

1. We are 90% confident that the interval from 3.2 to 4.0 mg/L contains the true mean nitrate concentration for the river locations of interest. (correct answer)
2. There is a 90% probability that the true mean nitrate concentration is between 3.2 and 4.0 mg/L.
3. About 90% of nitrate measurements at individual locations are between 3.2 and 4.0 mg/L.
4. If many samples of 30 locations were taken, 90% of the resulting intervals would contain the sample mean from this study.
5. If many samples of 30 locations were taken, 90% of the time the population mean would vary and land inside (3.2, 4.0).

Explanation: This question assesses understanding of confidence interval interpretation. The correct interpretation (A) states we are 90% confident the interval contains the true mean nitrate concentration. Choice B incorrectly assigns probability to the parameter after interval calculation. Choice C wrongly applies the interval to individual location measurements rather than the mean. Choice D nonsensically suggests intervals would contain a specific sample mean. Choice E incorrectly implies the population mean varies between samples. Remember: the population mean is fixed but unknown; our confidence is in the method that produced the interval, not in probability statements about the parameter.

Question 16

1. There is a 99% chance that a randomly selected household uses between 280 and 340 gallons of water per day.
2. We are 99% confident that the true mean daily water usage for all households in the city is between 280 and 340 gallons. (correct answer)
3. The probability that the true mean daily water usage is between 280 and 340 gallons is 0.99.
4. About 99% of the households in the city have daily water usage between 280 and 340 gallons.
5. If the engineer repeatedly samples 60 households, 99% of the time the sample mean will fall between 280 and 340 gallons.

Explanation: This problem tests confidence interval interpretation for mean water usage. The correct answer (B) states we are 99% confident the true mean daily water usage is between 280 and 340 gallons. Choice A incorrectly applies the interval to individual households rather than the mean. Choice C wrongly assigns probability to the fixed population parameter. Choice D confuses the interval for the mean with individual household values. Choice E misinterprets the interval as capturing sample means from future samples. Remember that confidence intervals estimate population parameters, not individual values or future sample statistics.

Question 17

1. There is a 95% probability that the population mean sodium content is between 820 mg and 860 mg.
2. If many random samples of 40 cans were taken and 95% confidence intervals were constructed, about 95% of those intervals would contain the true population mean sodium content. (correct answer)
3. About 95% of the sodium contents of individual cans are between 820 mg and 860 mg.
4. There is a 95% probability that the sample mean sodium content is between 820 mg and 860 mg.
5. Exactly 95% of all possible sample means from samples of size 40 fall between 820 mg and 860 mg.

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, about 95% of the resulting confidence intervals would contain the true population mean. Choice A incorrectly treats the population mean as a random variable with probability - the true mean is fixed, not random. Choice C confuses the confidence interval for the mean with a prediction interval for individual values. Choice D incorrectly applies probability to the sample mean, which we already observed. Choice E misinterprets the interval as capturing sample means rather than containing the population mean. Remember: confidence intervals describe the reliability of our estimation procedure, not probabilities about parameters.

Question 18

1. If the teacher measured all plants in the greenhouse, 95% of the heights would be between 42.1 and 47.9 cm.
2. There is a 95% probability that the interval $(42.1,\ 47.9)$ contains the sample mean height.
3. We are 95% confident that the true mean height of all plants of this species in the greenhouse is between 42.1 and 47.9 cm. (correct answer)
4. About 95% of the time, the true mean height will change to fall between 42.1 and 47.9 cm.
5. There is a 95% chance that the true mean height is exactly 45.0 cm, the midpoint of the interval.

Explanation: This question involves interpreting a 95% CI for mean plant height. The correct answer (C) properly states we are 95% confident the true mean height is between 42.1 and 47.9 cm. Choice A incorrectly applies the interval to individual plants. Choice B wrongly suggests the sample mean is random after data collection. Choice D nonsensically suggests the population parameter changes. Choice E incorrectly focuses on the midpoint with a specific probability. A confidence interval gives us a range of plausible values for the unknown population mean based on our sample.

Question 19

1. If we repeatedly took random samples of 100 workers and built 95% confidence intervals, about 95% of those intervals would contain the true mean commute time for all workers in the city. (correct answer)
2. There is a 95% chance that the true mean commute time is between 27.4 and 31.8 minutes.
3. 95% of all workers in the city have commute times between 27.4 and 31.8 minutes.
4. About 95% of random samples of 100 workers will have sample means between 27.4 and 31.8 minutes.
5. Because the confidence interval is $(27.4, 31.8)$, the population mean commute time must be 29.6 minutes.

Explanation: This question evaluates proper interpretation of confidence intervals in terms of repeated sampling. The correct answer (A) accurately describes that if we repeatedly sampled and built intervals, about 95% would contain the true mean commute time. Choice B incorrectly assigns probability to the parameter after interval construction. Choice C wrongly interprets the interval as describing individual commute times. Choice D misunderstands what the interval represents about sample means. Choice E incorrectly assumes the parameter equals the interval midpoint. This interpretation emphasizes the frequentist view: confidence describes the long-run behavior of the interval construction method, not probability about a specific interval.

Question 20

1. There is a 95% chance that the population mean homework time is between 52.1 and 68.7 minutes.
2. We are 95% confident that the interval from 52.1 to 68.7 minutes captures the true population mean homework time. (correct answer)
3. About 95% of all students at the school spend between 52.1 and 68.7 minutes per day on homework.
4. If many random samples of 40 students were taken, 95% of the sample means would fall between 52.1 and 68.7 minutes.
5. If the study were repeated many times, 95% of the time the population mean would fall between 52.1 and 68.7 minutes.

Explanation: This question tests understanding of confidence interval interpretation for a population mean. The correct interpretation (B) states that we are 95% confident the interval captures the true population mean homework time. Choice A incorrectly treats the confidence level as a probability about the parameter itself. Choice C misinterprets the interval as describing individual student values rather than the mean. Choice D incorrectly describes a sampling distribution of sample means. Choice E wrongly suggests the population mean changes between studies. Remember: a confidence interval provides a range of plausible values for the population parameter based on our sample data, and the confidence level describes the long-run success rate of the method.