AP Statistics Quiz: Justifying Claims Population Mean Confidence Interval

What this quiz covers

This quiz focuses on Justifying Claims Population Mean Confidence Interval, 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. Yes, because 90% of all boxes contain between 15.6 oz and 15.9 oz.
2. No, because $16.0$ is not in the interval $(15.6,\ 15.9)$. (correct answer)
3. Yes, because the interval is based on a random sample.
4. Yes, because 90% confidence means there is a 90% chance the mean is 16.0 oz.
5. No, because the interval must include 16.0 oz to be valid.

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.

Question 2

1. Yes, because the interval is close to 45 minutes, so the claim is supported.
2. No, because 45 is not in the interval, so the claim is not supported. (correct answer)
3. Yes, because 95% confidence means the true mean is definitely between 41 and 44 minutes.
4. Yes, because 95% of delivery times are between 41 and 44 minutes.
5. No, because the confidence interval is based on a sample, so it can never be used to assess a claim.

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.

Question 3

1. No, because 3.10 is not exactly equal to either endpoint of the interval.
2. Yes, because 90% confidence means there is a 90% chance the true mean equals 3.10.
3. Yes, because 3.10 is in the interval, so the claim is plausible at the 90% confidence level. (correct answer)
4. No, because 90% confidence intervals are only valid for proportions, not means.
5. 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.

Question 4

1. Yes, because 95% of students spend between 84 and 96 minutes on homework.
2. No, because the interval does not contain 90, so the claim is not supported.
3. Yes, because 90 is in the interval, so the claim is plausible at the 95% confidence level. (correct answer)
4. No, because a 95% confidence interval means there is a 95% chance that $\mu$ is not 90.
5. Yes, because the interval guarantees the true mean is between 84 and 96 minutes.

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.

Question 5

1. Yes, because 90% of large cups contain between 15.6 and 15.9 oz.
2. No, because 16 is not in the interval, so the claim is not supported. (correct answer)
3. Yes, because a 90% confidence level means the true mean must equal 16 oz.
4. Yes, because the sample mean must have been 16 oz.
5. 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.

Question 6

1. No, because 92% confidence means there is an 8% chance the interval is correct.
2. Yes, because 10,000 is in the interval, so the claim is plausible at the 92% confidence level. (correct answer)
3. No, because the interval is too wide to support any claim.
4. Yes, because 92% of users take between 9,700 and 10,300 steps daily.
5. No, because the interval includes 10,300, which contradicts the claim of 10,000.

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.

Question 7

1. Yes, because 95% of users take between 9,450 and 9,980 steps per day.
2. No, because $10{,}000$ is not in $(9{,}450,\,9{,}980)$, so the claim is not plausible at the 95% confidence level. (correct answer)
3. Yes, because 95% confidence means there is a 95% chance the true mean is 10,000.
4. Yes, because the interval is close to 10,000, so it must include 10,000 in repeated samples.
5. No, because a confidence interval cannot be used unless the population mean is known.

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.

Question 8

1. No, because a 99% confidence interval means the mean cannot be exactly 30 hours.
2. Yes, because $30$ is in $(29.2,\,30.8)$, so the claim is plausible at the 99% confidence level. (correct answer)
3. No, because 99% confidence means there is a 1% chance that $\mu$ is between 29.2 and 30.8.
4. Yes, because 99% of individual batteries last between 29.2 and 30.8 hours.
5. Yes, because the confidence interval proves the manufacturer's claim is true.

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.

Question 9

1. No, because 95% confidence means there is a 95% chance the mean is greater than 25 minutes.
2. Yes, because 95% of residents have commute times between 25 and 29 minutes.
3. No, because the interval starts at 25, so 25 is excluded.
4. Yes, because $25$ is included in the interval $(25,\,29)$, so the claim is plausible at the 95% confidence level. (correct answer)
5. No, because a confidence interval cannot be used to evaluate a claim about a mean.

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.

Question 10

1. Yes, because 98% of players are between 72.4 and 75.1 inches tall.
2. Yes, because 98% confidence means there is a 98% probability that $\mu=72$.
3. No, because $72$ is not in $(72.4,\,75.1)$, so the claim is not plausible at the 98% confidence level. (correct answer)
4. Yes, because the interval is close to 72, so it supports the claim.
5. No, because a higher confidence level always makes the interval exclude the true mean.

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.

Question 11

1. Yes, because 98% confidence means 98% of households use between 300 and 315 gallons per day.
2. No, because 320 is not in the interval, so the claim is not supported. (correct answer)
3. Yes, because 320 is greater than both endpoints, which indicates higher use.
4. Yes, because a higher confidence level always supports the company's claim.
5. No, because 98% confidence means there is a 2% chance that $\mu=320$.

Explanation: This question asks whether a claim that μ = 320 gallons per day is supported by a 98% confidence interval of (300, 315) gallons. The claimed value of 320 is NOT in the interval - it exceeds the upper bound of 315. This provides strong evidence against the claim at the 98% confidence level. Choice A misinterprets the interval as describing individual households. Choice C incorrectly suggests that being above the interval somehow supports the claim. Choice E misunderstands confidence levels. The key principle: when a claimed value falls outside a confidence interval, we have statistical evidence to reject that claim at the given confidence level.

Question 12

1. Yes, because 95% confidence means the true mean is 28 minutes with 95% probability.
2. Yes, because 28 is close to the interval, so the claim is supported.
3. No, because 28 is not in the interval, so the claim is not supported. (correct answer)
4. Yes, because 95% of commute times are between 29.1 and 32.4 minutes.
5. No, because a confidence interval can never be used to evaluate a claim about a mean.

Explanation: This question tests whether a claim that μ = 28 minutes is supported by a 95% confidence interval of (29.1, 32.4) minutes. The claimed value of 28 is NOT in the interval - it falls below the lower bound of 29.1. This means we have evidence against the claim at the 95% confidence level. Choice B incorrectly suggests that being "close" to the interval is sufficient. Choice D misinterprets the interval as describing individual commute times rather than the mean. Choice E is false - confidence intervals are specifically designed to evaluate claims about parameters. The key principle: a claim is supported only if the claimed value falls within the confidence interval.

Question 13

1. Yes, because 20 is in the interval, so the claim is plausible at the 99% confidence level. (correct answer)
2. No, because 99% confidence means there is a 99% chance $\mu=20$ exactly.
3. No, because the interval contains many values besides 20, so it cannot support the claim.
4. Yes, because 99% of individual phones have battery life between 19.2 and 20.8 hours.
5. No, because a higher confidence level makes the interval less accurate and unusable.

Explanation: This question evaluates whether a claim that μ = 20 hours is supported by a 99% confidence interval of (19.2, 20.8) hours. Since 20 falls within the interval, the claim is plausible at the 99% confidence level. Choice D incorrectly interprets the interval as describing individual phones rather than the population mean. Choice C is wrong because a confidence interval naturally contains many plausible values for the parameter - that's its purpose. Choice E incorrectly suggests higher confidence levels make intervals less useful. When evaluating claims using confidence intervals, check if the claimed value is inside (supported) or outside (not supported) the interval.

Question 14

1. Yes, because $42$ is contained in $(38,\,46)$, so the claim is plausible at the 99% confidence level. (correct answer)
2. No, because 99% confidence means the true mean is 42 minutes with 99% certainty, which is impossible.
3. No, because the interval includes values other than 42, so the mean cannot be 42.
4. Yes, because 99% of patients waited between 38 and 46 minutes.
5. No, because a higher confidence level makes the interval too wide to evaluate the claim.

Explanation: This question evaluates justifying mean claims via confidence intervals in AP Statistics. The hospital's claim of μ=42 minutes is supported since 42 is within the 99% confidence interval (38, 46), making it plausible. Distractor choice D confuses the interval with individual wait times. Choice C incorrectly assumes inclusion of other values disproves the claim. Mini-lesson: Confidence intervals offer ranges for the mean, with higher levels providing broader but more reliable estimates. Inclusion supports the claim's consistency with data.

Question 15

1. Yes, because 95% of students spend between 84 and 96 minutes on homework each night.
2. No, because the interval shows the mean must be between 84 and 96, so it cannot be exactly 90.
3. Yes, because $90$ is contained in the interval $(84,\,96)$, so the claim is plausible at the 95% confidence level. (correct answer)
4. No, because a 95% confidence interval means there is a 95% chance that $\mu$ is not 90.
5. Yes, because the confidence level guarantees the true mean is 90.

Explanation: 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.

Question 16

1. No, because 92% confidence is not high enough to support any claim.
2. Yes, because $720$ is contained in $(705,735)$. (correct answer)
3. Yes, because 92% of servings contain between 705 mg and 735 mg of sodium.
4. No, because the interval includes many values besides 720 mg.
5. Yes, because there is a 92% chance that $\mu=720$ mg exactly.

Explanation: This question tests whether a nutrition claim is supported by a confidence interval. The label claims μ = 720 mg of sodium, and the 92% confidence interval is (705, 735) mg. Since 720 falls within this interval, the claim is supported by the data. Choice C makes the common error of interpreting the confidence interval as describing individual serving values rather than the population mean. Choice E misunderstands confidence levels—92% confidence means that if we repeated this sampling process many times, 92% of the resulting intervals would contain the true mean, not that there's a 92% chance μ equals exactly 720. To evaluate claims using confidence intervals, simply check whether the claimed value falls inside the interval; if it does, the claim is plausible given the sample evidence.

Question 17

1. Yes, because 90% of soup servings have sodium between 692 mg and 715 mg.
2. No, because $680$ is not in $(692,\,715)$, so the claim is not plausible at the 90% confidence level. (correct answer)
3. Yes, because 90% confidence means there is a 90% chance that $\mu=680$.
4. No, because the interval is only 90% confident and therefore cannot be used to assess the claim.
5. Yes, because the sample mean must have been close to 680 mg even if 680 is not in the interval.

Explanation: This question tests the ability to justify claims about a population mean using a confidence interval in AP Statistics. The nutritionist's claim that μ=680 mg is not supported because 680 is not within the 90% confidence interval (692, 715), indicating it's not a plausible value at this confidence level. Choice A is a distractor that confuses the interval for the mean with the distribution of individual sodium levels, as confidence intervals estimate the mean, not individual values. Choice C incorrectly assigns the confidence level as a probability for a specific mean value, which misrepresents the interval's meaning. A mini-lesson on confidence intervals: they estimate a range of plausible values for the population mean, where the confidence level reflects the long-run success rate of the method in capturing the true mean across many samples. Therefore, if the claimed value is outside the interval, the claim is not supported by the data.

Question 18

1. Yes, because 95% confidence means 95% of commute times are between 29.1 and 33.4 minutes.
2. No, because $28$ is not contained in $(29.1,33.4)$. (correct answer)
3. Yes, because the interval is close to 28 minutes, so the claim is essentially correct.
4. Yes, because the sample was random, so the interval must include the true mean.
5. No, because 95% confidence means there is a 95% chance the true mean is outside the interval.

Explanation: This question asks whether a claim about mean commute time is supported by a confidence interval. The city planner claims μ = 28 minutes, but the 95% confidence interval is (29.1, 33.4) minutes. Since 28 is not contained in this interval, the claim is not supported—we have evidence that the true mean commute time is longer than claimed. Choice A misinterprets the confidence interval as describing individual commute times rather than the population mean. Choice C incorrectly suggests that being "close" to the interval is sufficient, but in hypothesis testing, we need the claimed value to be inside the interval. When a confidence interval excludes a claimed value, we have statistical evidence to reject that claim at the given confidence level.

Question 19

1. No, because 95% confidence means 95% of days have between 110 and 130 calls, not the mean.
2. Yes, because $120$ is in $(110,\,130)$, so the claim is plausible at the 95% confidence level. (correct answer)
3. No, because the interval includes values other than 120, so the mean cannot be 120.
4. Yes, because 95% confidence guarantees the true mean is exactly 120.
5. No, because the confidence interval is centered at 120 so it must be biased.

Explanation: This question tests the use of confidence intervals to justify population mean claims in AP Statistics. The company's claim of μ=120 calls is supported because 120 is within the 95% confidence interval (110, 130), indicating plausibility. Distractor choice A correctly notes the interval isn't for individual days but fails to recognize support for the mean claim. Choice D overinterprets the interval as guaranteeing exactness, which it does not. Mini-lesson: Confidence intervals provide a range of believable values for the mean, with the confidence level showing procedural reliability. Inclusion of the claimed value means the claim is consistent with the sample evidence.

Question 20

1. Yes, because the upper bound is greater than 25 cm.
2. No, because 95% confidence means 95% of fish are between 24.2 cm and 26.1 cm.
3. Yes, because 25 cm is inside the interval, so $\mu$ must be greater than 25 cm.
4. No, because the interval includes values less than 25 cm, so it does not support $\mu>25$. (correct answer)
5. Yes, because the confidence interval proves the claim is true.

Explanation: This question tests justifying inequality claims about a population mean using a confidence interval for μ. The biologist's claim that μ > 25 cm is not supported since the 95% confidence interval (24.2, 26.1) includes values below 25, allowing for μ possibly ≤ 25. A typical distractor is choice B, misapplying the interval to individual fish lengths. Choice C wrongly assumes inclusion implies the inequality holds. Mini-lesson: For μ > k, the interval must lie entirely above k for support; straddling k means the claim isn't confirmed. Focus on the mean, not variability in samples.