Confidence Interval for a Population Mean - AP Statistics

1. A city planner randomly samples 20 intersections and records the average wait time (seconds) at each intersection during rush hour. A one-sample $t$ interval produces a 98% confidence interval for the population mean wait time of $(41,\ 57)$ seconds. Which interpretation is correct?
   Answer: We are 98% confident that the interval from 41 to 57 seconds contains the true population mean wait time at intersections like these during rush hour. This question tests understanding of confidence interval interpretation for a mean. The correct interpretation (B) properly states we are 98% confident the interval contains the true population mean wait time. Choice A incorrectly assigns probability to the parameter after the interval is calculated. Choice C misapplies the interval to individual intersections rather than the mean. Choice D incorrectly describes where sample means would fall in repeated sampling. Choice E makes an unfounded claim about the exact value of the mean. Remember: high confidence doesn't imply the parameter equals the interval's midpoint; it only increases our confidence that the interval captures the true value.
2. A fitness coach wants to estimate the mean resting heart rate (in beats per minute) of all clients at a gym. She takes a random sample of 36 clients and computes a 90% confidence interval for the population mean resting heart rate: $(68.2, 72.9)$ bpm. Which interpretation is correct?
   Answer: We are 90% confident that the true mean resting heart rate for all gym clients is between 68.2 and 72.9 bpm. This question assesses proper interpretation of a confidence interval for mean resting heart rate. The correct answer (C) appropriately states we are 90% confident the true mean heart rate for all gym clients is between 68.2 and 72.9 bpm. Choice A incorrectly interprets the interval as describing individual heart rates. Choice B wrongly assigns probability to the parameter after interval calculation. Choice D misunderstands what the interval represents - it's about the mean, not all individual values in samples. Choice E makes a nonsensical statement about intervals from the same sample. Remember: confidence intervals estimate population parameters like means, with confidence describing the method's reliability.
3. 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?
   Answer: 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. 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.
4. 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?
   Answer: We are 90% confident that the true population mean daily rainfall in the region is between 0.18 and 0.34 inches. 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.
5. A botanist wants to estimate the mean height (in cm) of a certain species of sunflower in a large field. She randomly selects 20 sunflowers and measures their heights. A 85% confidence interval for the population mean height is $(142, 151)$ cm. Which interpretation is correct?
   Answer: We are 85% confident that the population mean height of this sunflower species in the field is between 142 cm and 151 cm. This question assesses confidence interval interpretation in a botanical context. The correct answer (B) states that we are 85% confident the population mean height is between 142 and 151 cm. Option A incorrectly assigns probability to the parameter after data collection. Option C wrongly interprets the interval as describing individual sunflower heights. Option D confuses future sample means with the confidence interval. Option E makes an incorrect statement about intervals containing the original sample mean. Key concept: lower confidence levels (85%) produce narrower intervals but maintain the same interpretation framework.
6. A gym manager randomly sampled 20 treadmills of the same model and recorded the maximum speed (in mph) each could reach. A 90% confidence interval for the population mean maximum speed was $(11.6, 12.4)$ mph. Which interpretation is correct?
   Answer: We are 90% confident that the true population mean maximum speed for this treadmill model is between 11.6 and 12.4 mph. This question tests interpreting a confidence interval for a population mean in AP Statistics. The correct choice, A, says we are 90% confident the true population mean maximum speed is between 11.6 and 12.4 mph. Choice C is a distractor that assigns a 90% probability to the mean being in the interval, misrepresenting the fixed mean. For a mini-lesson on mean confidence intervals, they use sample statistics to estimate the population parameter with a specified confidence level reflecting long-run performance. The construction accounts for sample size and variability. This helps differentiate from statements about individual treadmill speeds.
7. An environmental scientist randomly samples 30 locations along a river and measures nitrate concentration (mg/L). A one-sample $t$ interval gives a 90% confidence interval for the population mean nitrate concentration of $(3.2,\ 4.0)$ mg/L. Which interpretation is correct?
   Answer: 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. 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.
8. A university randomly samples 50 first-year students and records the number of hours they sleep per night. A 95% confidence interval for the population mean sleep time is reported as $(6.85,\ 7.43)$ hours. Which interpretation is correct?
   Answer: We are 95% confident that the true population mean sleep time is between 6.85 and 7.43 hours per night. This question evaluates proper confidence interval interpretation. The correct answer (B) states we are 95% confident the true population mean sleep time is in the given interval. Choice A incorrectly applies the interval to individual student sleep times rather than the mean. Choice C wrongly treats confidence as probability about the parameter. Choice D describes the sampling distribution of sample means, not confidence interval interpretation. Choice E makes a nonsensical statement about intervals containing a sample mean. Key concept: confidence describes the long-run success rate of the interval construction method, not probability about a specific interval or descriptions of individual values.
9. A biology teacher wants to estimate the mean height (in centimeters) of a certain species of plant in a large greenhouse. The teacher randomly selects 18 plants and measures their heights. A 95% confidence interval for the population mean plant height is reported as $(42.1,\ 47.9)$ cm. Which interpretation is correct?
   Answer: 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. 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.
10. A psychologist randomly samples 22 adults and measures their score on a stress scale. A one-sample $t$ interval is used, and a 92% confidence interval for the population mean stress score is $(18.3,\ 21.6)$. Which interpretation is correct?
    Answer: We are 92% confident that the interval from 18.3 to 21.6 contains the true population mean stress score for adults like those in the sampling frame. This question tests proper confidence interval interpretation for a population mean. The correct answer (A) appropriately states we are 92% confident the interval contains the true population mean stress score. Choice B incorrectly treats confidence as probability about the parameter. Choice C misapplies the interval to individual adult stress scores rather than the mean. Choice D wrongly describes where sample means would fall in repeated sampling. Choice E makes a nonsensical claim about the confidence level determining the parameter's location within the interval. Key concept: confidence describes our trust in the interval construction method, not probability statements about where the parameter lies.
11. A psychologist wants to estimate the mean score on a stress questionnaire for employees at a large company. A random sample of 45 employees is taken, and a 95% confidence interval for the population mean stress score is $(31.0, 35.8)$. Which interpretation is correct?
    Answer: If many random samples of 45 employees are taken and a 95% confidence interval is computed each time, about 95% of those intervals will contain the true population mean stress score. This question tests the frequentist interpretation of confidence intervals. The correct answer (B) properly describes that if many samples are taken and intervals computed, about 95% will contain the true population mean. Option A incorrectly interprets the interval as describing individual employee scores. Option C wrongly assigns probability to the parameter. Option D misinterprets the interval as describing individual values within samples. Option E incorrectly refers to the sample mean, which is known with certainty. This interpretation emphasizes the long-run behavior of the confidence interval procedure across repeated sampling.
12. 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?
    Answer: 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. 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.
13. A researcher is estimating the mean systolic blood pressure (in mmHg) of adults in a city. She takes a random sample of 64 adults and computes a 90% confidence interval for the population mean systolic blood pressure: (121, 127) mmHg. Which interpretation is correct?
    Answer: If the researcher repeats this sampling process many times, about 90% of the 90% confidence intervals constructed will contain the true population mean systolic blood pressure. This question assesses understanding of confidence interval interpretation for mean blood pressure. The correct answer (B) properly describes that if the sampling process were repeated many times, about 90% of the resulting confidence intervals would contain the true population mean. Choice A incorrectly treats the population mean as having probability. Choice C confuses the interval for the mean with a range for individual blood pressure values. Choice D misinterprets the interval as capturing future sample means. Choice E makes a correct statement about interval width but doesn't interpret what the interval means. The key insight is that confidence describes the long-run performance of the interval construction procedure.
14. 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?
    Answer: We are 98% confident that the true mean completion time for all students in the district is between 47.5 and 52.9 minutes. 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.
15. A coffee shop owner randomly samples 18 days and records the mean number of customers per day. Using a one-sample $t$ procedure, the owner reports a 90% confidence interval for the population mean number of customers per day of $(118,\ 134)$. Which interpretation is correct?
    Answer: We are 90% confident that the interval from 118 to 134 customers contains the true population mean number of customers per day. 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.
16. An environmental scientist estimates the mean nitrate concentration (in mg/L) in a river by taking a random sample of 25 water samples from the river on a particular day. A 90% confidence interval for the population mean nitrate concentration is $(3.8, 5.1)$ mg/L. Which interpretation is correct?
    Answer: We are 90% confident that the true mean nitrate concentration in the river on that day is between 3.8 and 5.1 mg/L. This question assesses proper interpretation of a confidence interval for mean nitrate concentration. The correct answer (A) properly states that we are 90% confident the true mean concentration is between 3.8 and 5.1 mg/L. Choice B incorrectly assigns probability to the parameter after the interval is calculated. Choice C wrongly interprets the interval as describing individual measurements rather than the mean. Choice D misunderstands what values fall in the interval. Choice E incorrectly describes the sampling distribution of sample means. Key concept: confidence intervals estimate population parameters (like means), not individual values, and the confidence level refers to the method's reliability, not probability about a specific interval.
17. To estimate the mean amount of time (in seconds) it takes a website to load for all users, a random sample of 60 load times was collected. A 99% confidence interval for the population mean load time was $(2.10, 2.55)$ seconds. Which interpretation is correct?
    Answer: We are 99% confident that the true population mean load time for all users is between 2.10 and 2.55 seconds. 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.
18. A school nurse took a random sample of 50 students and measured their resting heart rates (beats per minute). A 92% confidence interval for the population mean resting heart rate was $(71, 76)$ bpm. Which interpretation is correct?
    Answer: If the nurse repeatedly took random samples of 50 students and built a 92% confidence interval each time, about 92% of those intervals would contain the true population mean resting heart rate. This question assesses the interpretation of a confidence interval for a population mean in AP Statistics. Choice B is correct, explaining that if random samples of 50 students are repeatedly taken and 92% confidence intervals built, about 92% of those intervals would contain the true population mean resting heart rate. A typical distractor is choice A, which mistakenly applies probability to the population mean being in this specific interval. In a mini-lesson on confidence intervals for means, the interval is calculated as the sample mean ± t* (standard error), where the confidence level determines the critical value. This setup ensures that over many samples, the specified percentage of intervals cover the true mean. Understanding this repeated sampling perspective prevents confusion with individual data points or sample means.
19. 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?
    Answer: 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. 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.
20. 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?
    Answer: 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. 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.

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