AP Statistics Practice Quiz: Justifying a Claim About the Difference of Two Means 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 true difference in the mean test scores between the population of students at School A and School B is between 2.5 and 8.3 points.
B) The difference between the two sample mean test scores is between 2.5 and 8.3 points, 95% of the time.
C) There is a 95% probability that the true difference in the mean test scores for the two populations is between 2.5 and 8.3 points.
D) We are 95% confident that the mean score for the sample from School A is between 2.5 and 8.3 points higher than the mean score for the sample from School B.
Correct Answer: A
A correct interpretation of a confidence interval for the difference of two means must reference the populations from which the samples were drawn and state the confidence level for the range of plausible values for the true difference in population means.
A) There is convincing evidence that Fertilizer X results in a greater mean plant growth than Fertilizer Y.
B) There is convincing evidence that Fertilizer Y results in a greater mean plant growth than Fertilizer X.
C) Since the interval contains both negative and positive values, there is no convincing evidence of a difference in the mean plant growth between the two fertilizers.
D) The difference in the sample means must be exactly 1.1 cm, which is the center of the interval.
Correct Answer: C
The confidence interval provides a range of plausible values for the true difference in population means. Because the value 0 is included in the interval (-1.2, 3.4), it is plausible that there is no difference between the population means. Therefore, the interval does not support a claim of a difference.
A) The width of the interval will tend to increase.
B) The width of the interval will tend to decrease.
C) The center of the interval will shift significantly to the right.
D) The width of the interval will not be affected by sample size.
Correct Answer: B
The width of the confidence interval for the difference of two means tends to decrease as sample sizes increase. Larger samples provide a more precise estimate of the true difference in population means.
A) 95% of the sample differences will fall within any single researcher's calculated interval.
B) There is a 95% probability that the true difference in population means is captured by one specific, calculated interval.
C) If the process of sampling and constructing intervals is repeated many times, approximately 95% of the resulting intervals will capture the true difference in population means.
D) 95% of the data points from the two samples are located between the lower and upper bounds of the interval.
Correct Answer: C
The confidence level, C%, indicates that in repeated sampling, approximately C% of the confidence intervals created will capture the true parameter. It refers to the reliability of the method, not the probability associated with a single, specific interval.
A) There is no statistically significant difference in the mean commute times between the two routes.
B) The mean commute time for Route 1 is plausibly greater than the mean commute time for Route 2.
C) There is convincing evidence that the mean commute time for Route 1 is less than the mean commute time for Route 2.
D) The samples provide insufficient evidence to make a claim about the population means.
Correct Answer: C
The confidence interval provides a range of plausible values for the true difference in population means (μ1 - μ2). Since all values in the interval are negative, this supports the claim that μ1 - μ2 < 0, which implies that the mean for Route 1 is less than the mean for Route 2.
A) A statement about the probability that the interval contains the difference in sample means.
B) A reference to the populations of trees from which the samples were drawn.
C) A claim that 95% of all trees from Species A are taller than trees from Species B.
D) A conclusion that the sampling method was not random.
Correct Answer: B
A correct interpretation for a confidence interval for the difference of two means must make an inference about the populations, not just the samples used to construct the interval. Therefore, referencing the populations is essential.
A) The interval from Study 2 would tend to be wider.
B) The interval from Study 2 would tend to be narrower.
C) The widths of the intervals would tend to be the same.
D) The interval from Study 2 would be centered at a larger value.
Correct Answer: B
The width of a confidence interval for the difference of two means is affected by sample size. As sample sizes increase, the width of the interval tends to decrease. Since Study 2 has larger sample sizes, its interval is expected to be narrower.
A) plausible values for the difference in the sample mean salaries.
B) plausible values for the true difference in the population mean salaries.
C) the exact range of all possible differences in individual salaries.
D) proof that every accountant earns between $5,000 and $12,000 more than every teacher.
Correct Answer: B
A confidence interval provides a range of plausible values for a population parameter. In this case, the parameter is the true difference in the population mean salaries, not the difference in sample means or individual salaries.
A) Online shoppers are significantly more satisfied than in-store shoppers.
B) In-store shoppers are significantly more satisfied than online shoppers.
C) The interval provides convincing evidence of a difference in mean satisfaction scores.
D) The interval does not provide convincing evidence of a difference in mean satisfaction scores.
Correct Answer: D
To justify a claim of a difference between two population means, the confidence interval for the difference must not contain 0. Since this interval, (-0.4, 0.7), contains 0, it is a plausible value for the true difference. Therefore, there is no convincing evidence of a difference.
A) The interpretation should be about the sample means, not the population means.
B) The confidence level refers to the long-run success rate of the method in repeated sampling, not the probability for a single, calculated interval.
C) The interval (1.5, 4.5) actually means there is no difference between the means.
D) The probability is either 0 or 1, but the student has not specified which.
Correct Answer: B
This is a common misconception. The true difference in population means is a fixed value. A specific, calculated confidence interval either contains this value or it does not. The 90% confidence level refers to the fact that if we were to repeat the sampling process many times, approximately 90% of the intervals we construct would capture the true difference.
A) Decrease the confidence level from 95% to 90%.
B) Increase the sample sizes of both groups.
C) Hope that the sample means from the next study are closer together.
D) Use a different formula for the confidence interval.
Correct Answer: B
The width of the confidence interval for the difference of two means tends to decrease as sample sizes increase. While decreasing the confidence level also narrows the interval, increasing sample size is the standard method for improving the precision of the estimate.
A) It is plausible that the old fertilizer has a higher mean yield than the new fertilizer.
B) There is no evidence of a difference in mean crop yield between the two fertilizers.
C) There is convincing evidence that the mean crop yield for the new fertilizer is greater than for the old fertilizer.
D) The difference in the sample mean yields was exactly 22.85 bushels per acre.
Correct Answer: C
The confidence interval provides a range of plausible values for the true difference in population means. Since the entire interval (10.2, 35.5) consists of positive values, 0 is not a plausible value. This provides convincing evidence that the difference (μ_New - μ_Old) is positive, meaning the new fertilizer has a greater mean yield.
A) We are 99% confident that the true difference in the mean puzzle completion time for the populations these groups were sampled from is between 1.2 and 5.8 minutes.
B) In 99% of future samples, the difference in the sample means will be between 1.2 and 5.8 minutes.
C) There is a 99% chance that the true mean time for Group A is exactly 3.5 minutes longer than for Group B.
D) Based on these samples, we can conclude there is no difference in the population mean times, as the interval is relatively narrow.
Correct Answer: A
This is a standard, correct interpretation. It correctly references the confidence level (99%), the parameter of interest (the true difference in population means), and provides the range of plausible values based on the samples.
A) Larger samples guarantee that the sample means will be closer to the population means.
B) The width of the confidence interval for the difference of two means tends to decrease as sample sizes increase.
C) Larger samples will always result in a confidence interval that does not contain zero.
D) Increasing sample size increases the standard deviation, which makes the interval narrower.
Correct Answer: B
The formula for the confidence interval width is directly related to the standard error, which decreases as sample size increases. Therefore, increasing sample sizes leads to a smaller standard error and, consequently, a narrower, more precise confidence interval.
A) For any given sample, there is a 90% probability that the true difference of population means is inside the calculated interval.
B) 90% of the data from the two samples combined will fall within the calculated interval.
C) In repeated random sampling, approximately 90% of the confidence intervals created by this method will capture the true difference of population means.
D) The difference between the sample means will be within the interval 90% of the time.
Correct Answer: C
The confidence level describes the long-run performance of the statistical method. It means that if we were to take many random samples and construct an interval for each one, we would expect about 90% of those intervals to successfully capture the true, unknown difference in the population means.
A) The interval provides a set of plausible values for the true difference in mean cholesterol reduction. Since all these values are positive, it supports the claim that Drug X has a greater mean reduction than Drug Y.
B) The interval is too wide to make a conclusion, so a new study with larger sample sizes must be conducted.
C) The interval supports the claim that Drug Y is more effective because the lower bound of 2.1 is close to zero.
D) In repeated sampling, 95% of such intervals will show that Drug X is more effective than Drug Y.
Correct Answer: A
This conclusion correctly identifies the interval as a set of plausible values for the parameter and uses the fact that all values are positive (0 is not in the interval) to justify the claim that μ_X is greater than μ_Y. It correctly links the concept of plausible values to justifying a claim.