Justifying Claims: Population Mean, Confidence Interval
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AP Statistics › Justifying Claims: Population Mean, Confidence Interval
A cereal company claims the population mean amount of cereal in its boxes is $\mu=16.0$ oz. A quality-control team takes a random sample of boxes and reports a 90% confidence interval for $\mu$: $(15.6,\ 15.9)$. Is the claim supported by the CI?
Yes, because 90% confidence means there is a 90% chance the mean is 16.0 oz.
No, because the interval must include 16.0 oz to be valid.
No, because $16.0$ is not in the interval $(15.6,\ 15.9)$.
Yes, because the interval is based on a random sample.
Yes, because 90% of all boxes contain between 15.6 oz and 15.9 oz.
Explanation
This question evaluates the ability to justify claims about a population mean using a confidence interval for μ. The cereal company's claim that μ = 16.0 oz is not supported because 16.0 is not contained in the 90% confidence interval (15.6, 15.9), indicating it's not a plausible value. A frequent distractor is choice A, which wrongly assumes the interval describes the distribution of individual box weights instead of the mean. Choice D also misinterprets the confidence level as a probability that the mean equals the claimed value. Mini-lesson on confidence intervals: they provide a range where the true population mean is likely to lie; for equality claims, support exists only if the exact value is inside the interval. If the value is outside, the claim is contradicted by the data.
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?
Yes, because 95% confidence means the true mean is definitely between 41 and 44 minutes.
No, because 45 is not in the interval, so the claim is not supported.
Yes, because the interval is close to 45 minutes, so the claim is supported.
No, because the confidence interval is based on a sample, so it can never be used to assess a claim.
Yes, because 95% of delivery times are between 41 and 44 minutes.
Explanation
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 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?
No, because 90% confidence intervals are only valid for proportions, not means.
Yes, because 3.10 is in the interval, so the claim is plausible at the 90% confidence level.
Yes, because 90% confidence means there is a 90% chance the true mean equals 3.10.
No, because 3.10 is not exactly equal to either endpoint of the interval.
No, because the interval includes values not equal to 3.10, so the claim must be wrong.
Explanation
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 school district claims the mean time (in minutes) that students spend on homework each night is $\mu=90$. A random sample of students was used to construct a 95% confidence interval for $\mu$, resulting in $(84,\ 96)$ minutes. Is the claim supported by the confidence interval?
Yes, because 90 is in the interval, so the claim is plausible at the 95% confidence level.
Yes, because the interval guarantees the true mean is between 84 and 96 minutes.
No, because the interval does not contain 90, so the claim is not supported.
Yes, because 95% of students spend between 84 and 96 minutes on homework.
No, because a 95% confidence interval means there is a 95% chance that $\mu$ is not 90.
Explanation
This question tests whether you can use a confidence interval to evaluate a claim about a population mean. The claim states that μ = 90 minutes, and the 95% confidence interval is (84, 96) minutes. Since 90 falls within this interval, the claim is plausible - we cannot reject it at the 95% confidence level. Choice A incorrectly interprets the interval as describing individual students rather than the mean. Choice D misunderstands confidence levels - a 95% CI doesn't mean there's a 95% chance μ is not 90. When a claimed value falls inside a confidence interval, it means that value is consistent with our sample data at the given confidence level.
A coffee shop claims the mean amount of coffee dispensed per “large” cup is $\mu=16$ oz. A manager 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?
No, because 16 is not in the interval, so the claim is not supported.
Yes, because a 90% confidence level means the true mean must equal 16 oz.
Yes, because 90% of large cups contain between 15.6 and 15.9 oz.
Yes, because the sample mean must have been 16 oz.
No, because a 90% confidence interval is too low to evaluate any claim.
Explanation
This question asks whether a claim about the mean coffee amount (μ = 16 oz) is supported by a 90% confidence interval of (15.6, 15.9) oz. The key observation is that 16 is NOT contained within the interval (15.6, 15.9). When a claimed value falls outside the confidence interval, we have evidence against that claim at the given confidence level. Choice A incorrectly interprets the interval as describing individual cups rather than the mean. Choice C wrongly suggests that confidence levels determine exact values. The correct interpretation is that since 16 oz lies outside our 90% confidence interval, the claim is 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?
Yes, because 92% of users take between 9,700 and 10,300 steps daily.
No, because the interval includes 10,300, which contradicts the claim of 10,000.
No, because the interval is too wide to support any claim.
Yes, because 10,000 is in the interval, so the claim is plausible at the 92% confidence level.
No, because 92% confidence means there is an 8% chance the interval is correct.
Explanation
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 fitness app claims the mean number of steps users take per day is $\mu=10{,}000$. From a random sample of users, a 95% confidence interval for $\mu$ is $(9{,}450,,9{,}980)$. Is the claim supported by the confidence interval?
Yes, because 95% of users take between 9,450 and 9,980 steps per day.
No, because a confidence interval cannot be used unless the population mean is known.
Yes, because the interval is close to 10,000, so it must include 10,000 in repeated samples.
Yes, because 95% confidence means there is a 95% chance the true mean is 10,000.
No, because $10{,}000$ is not in $(9{,}450,,9{,}980)$, so the claim is not plausible at the 95% confidence level.
Explanation
This question assesses confidence interval use for mean claims in AP Statistics. The app's claim of μ=10,000 steps is not supported because 10,000 exceeds the 95% confidence interval (9,450, 9,980), indicating implausibility. Choice A mistakenly applies the interval to individual users' steps, not the mean. Choice C misuses the confidence level as probability for the claim. Mini-lesson: Intervals capture likely mean values, with confidence denoting long-term capture rate. Exclusion means the claim is inconsistent with the sample.
A manufacturer claims the mean lifetime of a certain battery is $\mu=30$ hours. A random sample of batteries is tested and a 99% confidence interval for $\mu$ is reported as $(29.2,,30.8)$. Is the claim supported by the confidence interval?
No, because 99% confidence means there is a 1% chance that $\mu$ is between 29.2 and 30.8.
Yes, because 99% of individual batteries last between 29.2 and 30.8 hours.
Yes, because $30$ is in $(29.2,,30.8)$, so the claim is plausible at the 99% confidence level.
Yes, because the confidence interval proves the manufacturer’s claim is true.
No, because a 99% confidence interval means the mean cannot be exactly 30 hours.
Explanation
This question evaluates the skill of using confidence intervals to justify claims about a population mean in AP Statistics. The manufacturer's claim of μ=30 hours is supported since 30 is inside the 99% confidence interval (29.2, 30.8), rendering it plausible at this high confidence level. Distractor choice D errs by suggesting the interval captures 99% of individual battery lifetimes, whereas it actually estimates the mean lifetime. Choice E overstates the interval as proof of the claim, but intervals provide plausibility, not certainty. Mini-lesson: Confidence intervals offer a range of values likely to include the true population mean, with the confidence level denoting the expected proportion of intervals containing the mean in repeated sampling. When the claimed mean is within the interval, the claim aligns with the data; otherwise, it's questioned.
A city official claims the mean commute time for residents is $\mu=25$ minutes. A random sample of residents yields a 95% confidence interval for $\mu$ of $(25,,29)$. Is the claim supported by the confidence interval?
No, because 95% confidence means there is a 95% chance the mean is greater than 25 minutes.
Yes, because $25$ is included in the interval $(25,,29)$, so the claim is plausible at the 95% confidence level.
No, because a confidence interval cannot be used to evaluate a claim about a mean.
No, because the interval starts at 25, so 25 is excluded.
Yes, because 95% of residents have commute times between 25 and 29 minutes.
Explanation
This question focuses on justifying population mean claims with confidence intervals in AP Statistics. The official's claim that μ=25 minutes is supported because 25 is at the boundary of the 95% confidence interval (25, 29), and in statistical practice, boundary values are considered included for plausibility. Distractor choice B wrongly claims the interval describes 95% of individual commute times, but it's for the mean. Choice C misinterprets the open parenthesis as strictly excluding 25, though confidence intervals are conventionally treated as closed for evaluation purposes. Mini-lesson: Confidence intervals construct a plausible range for the population mean based on sample data, with the level indicating method reliability over many samples. A claim is supported if its value lies within or at the boundary of the interval, suggesting consistency with the observed data.
A coach claims the mean height of players on a team is $\mu=72$ inches. A random sample of players gives a 98% confidence interval for $\mu$ of $(72.4,,75.1)$. Is the claim supported by the confidence interval?
No, because $72$ is not in $(72.4,,75.1)$, so the claim is not plausible at the 98% confidence level.
Yes, because 98% confidence means there is a 98% probability that $\mu=72$.
Yes, because 98% of players are between 72.4 and 75.1 inches tall.
No, because a higher confidence level always makes the interval exclude the true mean.
Yes, because the interval is close to 72, so it supports the claim.
Explanation
This question examines justifying mean claims with confidence intervals in AP Statistics. The coach's claim of μ=72 inches is not supported since 72 is below the 98% confidence interval (72.4, 75.1), deeming it implausible. Choice A distracts by misinterpreting the interval as for individual heights rather than the mean. Choice B errs in assigning probability to a specific mean value. Mini-lesson: Confidence intervals bracket plausible mean values, with higher levels widening the range for greater assurance. Non-inclusion suggests the claim doesn't align with the data at that level.