Justifying Claims: Population Mean, Confidence Interval - AP Statistics

A school district claims that the mean time (in minutes) students spend on homework each night is $\mu=90$ . A random sample of students is used to construct a 95% confidence interval for $\mu$ , resulting in $(84,\,96)$ . Is the claim supported by the confidence interval?
Answer: Yes, because $90$ is contained in the interval $(84,\,96)$ , so the claim is plausible at the 95% confidence level. This question assesses the skill of justifying claims about a population mean using a confidence interval in AP Statistics. The school's claim that the mean homework time is μ=90 minutes is supported because 90 is contained within the 95% confidence interval (84, 96), making it a plausible value for the true mean at this confidence level. A common distractor, like choice A, mistakenly interprets the interval as capturing 95% of individual students' times, but confidence intervals pertain to the population mean, not individual observations. Another distractor, choice D, misinterprets the confidence level as the probability that the mean is not a specific value, which is incorrect. In a mini-lesson on using confidence intervals: they provide a range where the true population mean is likely to lie, with the confidence level indicating the proportion of such intervals that would contain the true mean if repeated samples were taken. Thus, if a claimed mean falls inside the interval, the data does not contradict the claim; if outside, it suggests the claim is implausible.
A professor claims the mean score on an exam is $\mu=78$ . A random sample of exam scores is used to construct a 90% confidence interval for $\mu$ of $(76,\,80)$ . Is the claim supported by the confidence interval?
Answer: Yes, because $78$ is in $(76,\,80)$ , so the claim is plausible at the 90% confidence level. This question targets the skill of using confidence intervals for population mean claims in AP Statistics. The professor's claim of μ=78 is supported as 78 falls inside the 90% confidence interval (76, 80), supporting plausibility. Distractor choice D confuses the interval with individual scores' distribution. Choice E overclaims proof, as intervals indicate likelihood, not certainty. Mini-lesson: Constructed from samples, confidence intervals estimate mean ranges, with levels reflecting method consistency. Claims within the interval are deemed reasonable based on the evidence.
A farmer claims the mean weight of apples from an orchard is $\mu=170$ grams. A random sample of apples is weighed and a 92% confidence interval for the population mean apple weight is $(171\text{ g},\ 176\text{ g})$ . Is the claim supported by the confidence interval?
Answer: No, because 170 g is not in the interval, so $\mu=170$ g is not plausible at the 92% confidence level. This question tests whether a farmer's claim about mean apple weight is supported by a confidence interval. The claim is μ = 170 grams, but the 92% confidence interval is (171 g, 176 g). Since 170 grams is not contained within this interval, it is not a plausible value for the population mean at the 92% confidence level, so the claim is not supported. Choice A misinterprets the interval as describing individual apple weights. Choice C incorrectly accepts any value below the interval. Choice D misunderstands confidence level as a probability about the parameter. Choice E is completely wrong - confidence intervals apply to any parameter, including means. When evaluating claims, the key question is simple: is the claimed value inside or outside the confidence interval?
A researcher claims the mean reaction time for adults on a certain task is $\mu=250$ milliseconds. From a random sample of adults, a 90% confidence interval for the population mean reaction time is constructed as $(245\text{ ms},\ 255\text{ ms})$ . Is the claim supported by the confidence interval?
Answer: Yes, because 250 ms is in the interval, so $\mu=250$ ms is plausible at the 90% confidence level. This question tests whether a researcher's claim about mean reaction time is supported by a confidence interval. The claim is μ = 250 milliseconds, and the 90% confidence interval is (245 ms, 255 ms). Since 250 ms falls within this interval, it is a plausible value for the population mean at the 90% confidence level, supporting the claim. Choice A misinterprets confidence level as a probability about specific parameter values. Choice C incorrectly thinks the mean must be an endpoint. Choice D confuses the interval with individual reaction times. Choice E wrongly states confidence intervals prove exact values. When using confidence intervals to evaluate claims, remember they provide a range of plausible values - any value inside is supported, any value outside is not.
A delivery service claims the mean delivery time for a certain route is $\mu=45$ minutes. A random sample of deliveries produces a 95% confidence interval for $\mu$ of $(41,\ 44)$ minutes. Is the claim supported by the confidence interval?
Answer: No, because 45 is not in the interval, so the claim is not supported. This question asks whether a claim that μ = 45 minutes is supported by a 95% confidence interval of (41, 44) minutes. The claimed value of 45 is NOT in the interval - it exceeds the upper bound of 44. This means we have evidence against the claim at the 95% confidence level. Choice A vaguely suggests being "close" is sufficient, which is incorrect. Choice C misinterprets confidence intervals as providing certainty about the parameter's location. Choice D incorrectly describes individual delivery times rather than the mean. The fundamental rule: a claim is supported only if the claimed value falls within the confidence interval boundaries.
A city health department samples 60 households and estimates the mean weekly amount of trash produced per household. A 95% confidence interval for the population mean is $(24.1, 27.6)$ pounds. The director claims that the true mean weekly trash per household is 25 pounds. Is the claim supported by the confidence interval?
Answer: Yes, because 25 is inside the 95% confidence interval. This question tests whether you can use a confidence interval to evaluate a claim about a population mean. Since the 95% confidence interval is (24.1, 27.6) pounds and the claimed value of 25 pounds falls within this interval, the claim is supported. The key insight is that a confidence interval contains all plausible values for the population mean at the given confidence level. Choice B incorrectly interprets confidence level as a probability about the true mean, while choice C wrongly suggests the interval guarantees a specific value. When evaluating claims using confidence intervals, check if the claimed value is inside the interval - if yes, the claim is supported; if no, the claim is not supported.
A coffee shop owner records the preparation times (in seconds) for a random sample of 25 lattes and computes a 99% confidence interval for the population mean prep time: $(112, 128)$ . The owner claims the true mean prep time is 120 seconds. Is the claim supported by the confidence interval?
Answer: Yes, because 120 is within the 99% confidence interval. This question evaluates whether a claim about mean latte preparation time is supported by a confidence interval. The 99% confidence interval is (112, 128) seconds, and the claimed value is 120 seconds. Since 120 falls within this interval, the claim is supported by the data. Choice D incorrectly states that confidence intervals guarantee specific values, while choice E misinterprets the interval as describing individual values rather than the population mean. A confidence interval provides a range of plausible values for the population parameter based on sample data. To evaluate a claim, simply check if the claimed value is contained within the interval - containment means the claim is consistent with the data at the specified confidence level.
A gym claims the mean weight loss for members after a 12-week program is $\mu=10$ lb. A random sample of participants is used to compute an 80% confidence interval for $\mu$ of $(7.5,\ 9.8)$ lb. Is the claim supported by the confidence interval?
Answer: No, because $10$ is not contained in $(7.5,9.8)$ . This question asks whether a weight loss claim is supported by a confidence interval. The gym claims μ = 10 lb, but the 80% confidence interval is (7.5, 9.8) lb. Since 10 is not contained in this interval, the claim is not supported—we have evidence that the true mean weight loss is less than advertised. Choice A incorrectly interprets the confidence interval as describing individual participant results rather than the population mean. Choice C suggests that being "slightly below" is acceptable, but statistical inference requires the claimed value to be within the interval for support. When using confidence intervals for hypothesis testing, exclusion of the claimed value provides evidence against the claim, regardless of how close the value is to the interval boundaries.
A hospital administrator claims the mean waiting time in the emergency room is $\mu=38$ minutes. A random sample of patients yields a 97% confidence interval for the population mean waiting time of $(33\text{ min},\ 37\text{ min})$ . Is the claim supported by the confidence interval?
Answer: No, because 38 is not in the interval, so $\mu=38$ minutes is not plausible at the 97% confidence level. This question asks whether a hospital administrator's claim about mean ER waiting time is supported by a confidence interval. The claim is μ = 38 minutes, but the 97% confidence interval is (33 min, 37 min). Since 38 minutes is not contained within this interval, it is not a plausible value for the population mean at the 97% confidence level, so the claim is not supported. Choice A misinterprets the interval as describing individual patient wait times. Choice C incorrectly thinks high confidence guarantees closeness to a value. Choice D wrongly accepts values outside the interval. Choice E confuses confidence intervals with ranges of individual observations. A confidence interval represents uncertainty about the population mean - values outside the interval are implausible at the given confidence level.
A botanist randomly samples 35 plants of a species and estimates the mean height (in cm) at maturity. A 90% confidence interval for the population mean height is $(41.0, 44.7)$ . The botanist claims the true mean height is 40 cm. Is the claim supported by the confidence interval?
Answer: No, because 40 is not contained in the 90% confidence interval. This question asks whether a claim about mean plant height is supported by a confidence interval. The 90% confidence interval is (41.0, 44.7) cm, and the claimed value is 40 cm. Since 40 is less than the lower bound of 41.0, it falls outside the interval and the claim is not supported. Choice A incorrectly suggests that values "close" to the interval are supported, but confidence intervals have definite boundaries. Choice C misinterprets the 10% as meaning values outside must be true, when it actually refers to the chance the interval fails to capture the true mean. The key principle is that confidence intervals contain all plausible values for the population mean - values outside the interval, regardless of how close, are not supported by the data.
A university claims the mean GPA of its first-year students is $\mu=3.10$ . From a random sample of first-year students, a 90% confidence interval for $\mu$ is $(3.05,\ 3.15)$ . Is the claim supported by the confidence interval?
Answer: Yes, because 3.10 is in the interval, so the claim is plausible at the 90% confidence level. This question evaluates whether a claim that μ = 3.10 is supported by a 90% confidence interval of (3.05, 3.15). Since 3.10 falls within this interval, the claim is plausible at the 90% confidence level. Choice A incorrectly suggests the claimed value must equal an endpoint. Choice B misinterprets confidence levels - 90% confidence doesn't mean there's a 90% chance the mean equals exactly 3.10. Choice D is false - confidence intervals work for any parameter. Choice E misunderstands the purpose of confidence intervals, which naturally contain a range of plausible values. The correct interpretation: 3.10 is inside the interval, so the claim is consistent with the sample data.
A coffee shop advertises that the mean amount of coffee in its "16 oz" cups is $\mu=16.0$ oz. A quality-control team takes a random sample of cups and computes a 90% confidence interval for $\mu$ as $(15.6,\ 15.9)$ oz. Is the claim supported by the confidence interval?
Answer: No, because $16.0$ oz is not contained in $(15.6,15.9)$ . This question asks whether a claim about mean coffee volume is supported by a confidence interval. The coffee shop claims μ = 16.0 oz, but the 90% confidence interval is (15.6, 15.9) oz. Since 16.0 is not contained within this interval, the claim is not supported—we have statistical evidence that the true mean is less than advertised. Choice C makes a common error by interpreting the confidence interval as describing individual cup volumes rather than the population mean. A confidence interval gives us a range of plausible values for the population parameter, not a range for individual observations. When a claimed value falls outside the confidence interval, we have evidence to reject that claim at the corresponding confidence level.
A phone manufacturer claims the mean battery life for a certain model is $\mu=18$ hours. From a random sample of users, a 99% confidence interval for $\mu$ is computed as $(17.2,\ 18.9)$ hours. Is the claim supported by the confidence interval?
Answer: Yes, because $18$ is contained in $(17.2,18.9)$ . This question tests understanding of how to use confidence intervals to evaluate claims about population means. The manufacturer claims μ = 18 hours, and the 99% confidence interval is (17.2, 18.9) hours. Since 18 falls within this interval, the claim is supported by the data. Choice C incorrectly interprets the confidence interval as describing individual battery lives—this is a common misconception. The confidence interval describes plausible values for the population mean, not the range of individual values. Choice D misunderstands that we don't need to prove the mean is exactly 18; we only need to show that 18 is a plausible value given our data. When evaluating claims with confidence intervals, containment within the interval indicates the claim is consistent with the sample evidence.
A university surveys a random sample of 40 students and constructs a 90% confidence interval for the population mean number of hours slept per night: $(6.2, 7.0)$ hours. A researcher claims the true mean sleep time is 7.5 hours. Is the claim supported by the confidence interval?
Answer: No, because 7.5 is not in the 90% confidence interval. This question asks whether a claim about mean sleep time is supported by a confidence interval. The 90% confidence interval is (6.2, 7.0) hours, and the claimed value is 7.5 hours. Since 7.5 is outside (greater than) the upper bound of 7.0, the claim is not supported by the data. Choice C incorrectly suggests that values outside the interval are consistent with it, while choice E wrongly implies that any value could be the true mean. The fundamental principle is that a confidence interval contains the plausible values for the population parameter - values outside the interval are implausible at the given confidence level. Always check containment: if the claimed value is not in the interval, the claim is not supported.
A city planner claims the population mean commute time for city residents is less than 30 minutes. A random sample of residents is used to create a 99% confidence interval for the mean commute time $\mu$ : $(28.5,\ 31.2)$ . Is the claim supported by the CI?
Answer: No, because the interval includes values greater than 30 minutes, so it does not support $\mu<30$ . This question assesses justifying claims about a population mean with a confidence interval, specifically for an inequality claim about μ. The city planner's claim that μ < 30 minutes is not supported because the 99% confidence interval (28.5, 31.2) includes values greater than 30, meaning μ could plausibly be above 30. A common distractor is choice A, which focuses only on the lower bound without considering the full range. Choice B incorrectly applies the confidence level to individual commute times rather than the mean. Mini-lesson: For inequality claims like μ < k, the interval supports it only if all plausible values are less than k; if any exceed k, the claim lacks support. Confidence intervals estimate the mean, not individual observations.
A bookstore manager claims the mean amount spent per customer visit is $\mu=22$ dollars. A random sample of customers is used to compute an 88% confidence interval for the population mean spending per visit: $(19.5,\ 21.8)$ dollars. Is the claim supported by the confidence interval?
Answer: No, because $22 is not in the interval, so $\mu=22$ is not plausible at the 88% confidence level. This question evaluates whether a bookstore manager's claim about mean customer spending is supported by a confidence interval. The claim is μ = $22, but the 88% confidence interval is ($19.5, $21.8). Since $22 is not contained within this interval, it is not a plausible value for the population mean at the 88% confidence level, so the claim is not supported. Choice A misinterprets the interval as describing individual customer spending. Choice C incorrectly accepts the claim because values are close. Choice D wrongly thinks confidence level guarantees a specific value. Choice E is completely false - confidence intervals are specifically designed to evaluate claims about population parameters. The fundamental principle: if the claimed value is outside the confidence interval, we have evidence against that claim.
A gym owner claims the mean monthly membership fee paid by all members is $\mu=45$ dollars. A random sample of members is taken and a 98% confidence interval for the population mean monthly fee is computed: $(40.5,\ 49.2)$ dollars. Is the claim supported by the confidence interval?
Answer: Yes, because 45 is in the interval, so $\mu=45$ is plausible at the 98% confidence level. This question asks whether a gym owner's claim about mean membership fees is supported by a confidence interval. The claim is μ = $45, and the 98% confidence interval is ($40.5, $49.2). Since $45 falls within this interval, it is a plausible value for the population mean at the 98% confidence level, supporting the claim. Choice B misinterprets the confidence level as describing the proportion of members in the interval. Choice C incorrectly suggests wide intervals can't support claims. Choice D wrongly thinks confidence intervals prove exact values. Choice E incorrectly states the sample mean is needed to evaluate the claim. A confidence interval provides a range of plausible values for the population parameter - any value within the interval is supported at that confidence level.
An airline randomly samples 80 flights on a route and estimates the mean departure delay (in minutes). A 95% confidence interval for the population mean delay is $(8.5, 12.1)$ . A news report claims the true mean delay is 8 minutes. Is the claim supported by the confidence interval?
Answer: No, because 8 is not in the 95% confidence interval. This question tests whether a claim about mean flight delay is supported by a confidence interval. The 95% confidence interval is (8.5, 12.1) minutes, and the claimed value is 8 minutes. Since 8 is less than the lower bound of 8.5, it falls outside the interval, so the claim is not supported. Choice A incorrectly suggests that values "close to" the interval are supported, but confidence intervals have clear boundaries. Choice C misinterprets confidence level as a probability statement about specific values. The key principle is that confidence intervals contain all plausible values for the population mean at the given confidence level - values outside the interval, even if close, are not supported by the data.
A fitness tracker company claims the mean number of steps per day for its users is $\mu=10{,}000$ . A random sample of users yields a 92% confidence interval for $\mu$ of $(9{,}700,\ 10{,}300)$ . Is the claim supported by the confidence interval?
Answer: Yes, because 10,000 is in the interval, so the claim is plausible at the 92% confidence level. This question asks whether a claim that μ = 10,000 steps is supported by a 92% confidence interval of (9,700, 10,300). Since 10,000 falls within this interval, the claim is plausible at the 92% confidence level. Choice A misunderstands confidence levels - 92% confidence means we used a method that captures the true parameter 92% of the time, not that there's an 8% chance the interval is correct. Choice D incorrectly interprets the interval as describing individual users rather than the mean. Choice E is illogical - the interval containing values other than 10,000 doesn't contradict the claim. When a claimed value is inside the confidence interval, we say the claim is consistent with our data.
A manufacturer claims the population mean lifetime of its batteries is $\mu=12$ hours. A random sample of batteries is tested and a 95% confidence interval for $\mu$ is found to be $(11.3,\ 11.9)$ . Is the claim supported by the CI?
Answer: No, because $12$ is not in the interval $(11.3,\ 11.9)$ . This question assesses justifying population mean claims with a confidence interval for μ. The manufacturer's claim that μ = 12 hours is not supported as 12 is not in the 95% confidence interval (11.3, 11.9), excluding it from plausible values. A frequent distractor is choice B, incorrectly linking the interval to individual battery lifetimes. Choice E misstates the confidence level as the chance of error. Mini-lesson: Confidence intervals reject claims when the value is outside; they reflect sampling variability in estimating means, not guaranteeing exact values.

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