AP Statistics Practice Quiz: Justifying a Claim About a Population Mean Based on a Confidence Interval
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
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A) We are 95% confident that the interval from 150 mg to 250 mg captures the true mean daily caffeine intake for all office workers.
B) There is a 95% probability that the true mean daily caffeine intake for all office workers is between 150 mg and 250 mg.
C) 95% of office workers in the sample have a daily caffeine intake between 150 mg and 250 mg.
D) If we take many samples, 95% of them will have a sample mean between 150 mg and 250 mg.
Correct Answer: A
The correct interpretation states the confidence level and what the interval captures: the true population mean. It correctly references both the population ('all office workers') and the sample data used to construct the interval. Option B is incorrect because it implies the true mean is variable; the interval is what's variable. Option C incorrectly describes the sample data, not the population mean. Option D incorrectly describes the distribution of sample means.
A) The claim is not supported because the interval is not centered at 5 ounces.
B) The claim is supported because 5 ounces is a plausible value within the confidence interval.
C) The claim is not supported because the interval contains values less than 5 ounces.
D) The claim is proven to be true because 5 ounces is inside the interval.
Correct Answer: B
A confidence interval provides a range of plausible values for the population parameter. Since the company's claimed value of 5 ounces is contained within the interval (3.5, 5.5), it is considered a plausible value for the true mean increase. Therefore, the confidence interval supports the claim, it does not provide evidence against it.
A) Decrease the sample size.
B) Increase the sample size.
C) The width cannot be changed without changing the confidence level.
D) Recalculate the interval with the same data, as a narrower interval may result from random chance.
Correct Answer: B
The width of a confidence interval for a mean tends to decrease as the sample size increases. A larger sample provides more information about the population, leading to a more precise estimate and thus a narrower interval. Decreasing the sample size would make the interval wider.
A) The width of the interval will decrease.
B) The width of the interval will increase.
C) The width of the interval will remain the same, but its center will shift.
D) The effect on the width cannot be determined without knowing the sample mean.
Correct Answer: B
To be more confident that an interval captures the true population mean, we need to allow for a wider range of plausible values. Therefore, for a given sample, increasing the confidence level increases the critical value (t* or z*), which in turn increases the margin of error and the overall width of the confidence interval.
A) The specific interval calculated has a 95% probability of containing the true population mean.
B) 95% of the data in the population falls within the calculated interval.
C) If this sampling procedure were repeated many times, approximately 95% of the resulting confidence intervals would capture the true population mean.
D) The sample mean has a 95% chance of being equal to the population mean.
Correct Answer: C
The confidence level refers to the long-run success rate of the method used to create the interval. It does not describe the probability for any single, specific interval. A specific interval either contains the true mean or it does not. The 95% confidence is in the procedure, meaning that over many repetitions, about 95% of the intervals created will contain the true population mean.
A) We can be 99% confident that the diet plan causes a mean weight loss between 0.5 and 4.5 lbs.
B) The diet plan is ineffective because a mean difference of 0 is a plausible value.
C) 99% of participants lost between 0.5 and 4.5 lbs.
D) The diet plan is not effective because the interval is too wide.
Correct Answer: A
This is a matched pairs design. The confidence interval for the mean difference provides a range of plausible values for the true mean difference in the population. Since the entire interval (0.5, 4.5) is above zero, it suggests that a mean difference of zero is not a plausible value. This provides strong evidence that the diet plan is effective for weight loss.
A) The width would be one-fourth of the original width.
B) The width would be one-half of the original width.
C) The width would double.
D) The width would quadruple.
Correct Answer: B
The width of a confidence interval for a mean is proportional to 1/sqrt(n). If the sample size n is multiplied by 4, the new width will be proportional to 1/sqrt(4n) = 1/(2*sqrt(n)). This is one-half of the original proportion, 1/sqrt(n). Therefore, the width of the interval is expected to be halved.
A) Yes, because 85 is very close to the upper bound of 84.6.
B) Yes, because the interval is only a sample estimate.
C) No, because 85 is not contained within the interval of plausible values.
D) It is impossible to say without knowing the sample size.
Correct Answer: C
A confidence interval provides a range of plausible values for the population mean. Since the claimed value of 85 falls outside the calculated interval of (78.2, 84.6), the interval provides evidence that the claim is not correct. The value 85 is not considered a plausible value for the true mean score.
A) The probability that μ is in [10, 20] is 0.95.
B) The population mean μ is a fixed value; the interval [10, 20] either contains it or it does not.
C) The probability that μ is greater than 20 is 0.025.
D) If we took a new sample, the new 95% confidence interval would also be [10, 20].
Correct Answer: B
This question addresses a fundamental concept. The population mean (μ) is a fixed, unknown parameter. It does not vary. The confidence interval is what is random, as it is based on a random sample. Therefore, for any single interval that has been calculated, the true mean is either inside it or it is not. The 95% confidence refers to the method, not the probability of this specific interval.
A) The new shoe is significantly faster because the interval contains positive values.
B) The new shoe is significantly slower because the interval contains negative values.
C) There is no convincing evidence of a difference in mean times because 0 is a plausible value in the interval.
D) The experiment was flawed because the interval should not contain both negative and positive values.
Correct Answer: C
The confidence interval provides a range of plausible values for the true mean difference. A mean difference of 0 would indicate no change in speed between the shoes. Since 0 is contained within the interval (-0.05, 0.15), it is a plausible value. Therefore, we do not have convincing evidence to conclude that there is a difference in mean running times between the old and new shoes.
A) A 99% confidence level with a sample size of 100.
B) A 95% confidence level with a sample size of 100.
C) A 99% confidence level with a sample size of 50.
D) A 95% confidence level with a sample size of 50.
Correct Answer: C
The width of a confidence interval is affected by two main factors: the confidence level and the sample size. A higher confidence level leads to a wider interval. A smaller sample size leads to a wider interval. To get the widest possible interval, we need the highest confidence level (99%) and the smallest sample size (50).
A) The sample mean and the sample standard deviation.
B) The margin of error and the confidence level.
C) The sample from which the interval was constructed and the population to which we are generalizing.
D) The lower bound and the upper bound of the interval.
Correct Answer: C
A correct interpretation must be clear about the scope of the inference. It is based on data from a specific sample, and it provides a conclusion about a parameter of the larger population from which the sample was drawn. For example, 'Based on our sample of students, we are 95% confident the interval captures the true mean GPA of all students at the university.'
A) The batch fails to meet specification because the interval is not exactly centered at 50.
B) The batch fails to meet specification because the true mean is not known.
C) The batch meets specification because 50 grams is a plausible value contained within the interval.
D) The batch meets specification because the sample mean was 50 grams.
Correct Answer: C
The confidence interval provides a set of plausible values for the true population mean. Since the target value of 50 grams is inside the interval (49.8, 50.2), there is no statistical evidence to suggest the batch does not meet the specification. The sample mean is the center of the interval, which is 50 grams, but the justification comes from the target value being a plausible value in the interval.
A) The sample size must be doubled.
B) The sample size must be tripled.
C) The sample size must be quadrupled.
D) The sample size must be halved.
Correct Answer: C
The margin of error (ME) is proportional to 1/sqrt(n). To cut the margin of error in half (from 3 cm to 1.5 cm), we need to solve for the new sample size, n_new. ME_new = (1/2)ME_old. This means 1/sqrt(n_new) must be equal to (1/2) * (1/sqrt(n_old)). Squaring both sides gives 1/n_new = (1/4) * (1/n_old), which means n_new = 4 * n_old. The sample size must be quadrupled.
A) The sample size decreases.
B) The confidence level increases.
C) The sample standard deviation increases.
D) The sample size increases.
Correct Answer: D
The width of a confidence interval is directly related to the margin of error, which is proportional to 1/sqrt(n). As the sample size (n) increases, the denominator gets larger, making the margin of error smaller and the interval narrower. All other options would lead to a wider interval.
A) The claim of 350 gallons is plausible because the interval is close to 350.
B) The interval provides evidence that the mean usage is lower than the claimed 350 gallons.
C) The interval provides evidence that the mean usage is higher than the claimed 350 gallons.
D) The claim cannot be evaluated because the sample size is not given.
Correct Answer: B
The confidence interval represents the range of plausible values for the true mean household water usage. The claimed value of 350 gallons per day is not contained within the interval (335, 348). Since the entire interval is below 350, it provides statistical evidence to support the consumer group's belief that the true mean usage is lower than what the department claims.