AP Statistics Quiz: Confidence Intervals Difference Of Two Means

What this quiz covers

This quiz focuses on Confidence Intervals Difference Of Two Means, 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. We are 92% confident that $\mu_D-\mu_T$ is between 3 and 11 minutes.
2. We are 92% confident that public-transit commuters have a population mean commute time between 3 and 11 minutes longer than drivers. (correct answer)
3. There is a 92% chance that the true difference in means is exactly 7 minutes, the midpoint of the interval.
4. About 92% of all transit commuters take between 3 and 11 minutes longer than all drivers.
5. Since 0 is not in the interval, at least 92% of individual transit commute times exceed individual driving commute times.

Explanation: In AP Statistics, this problem involves a 92% confidence interval for μ_T - μ_D from 3 to 11 minutes. Choice B correctly interprets we are 92% confident transit commuters average 3 to 11 minutes longer than drivers in the population. The positive endpoints indicate transit likely takes longer on average. Choice D is a distractor, confusing the mean difference with comparisons of all individuals. Mini-lesson: Confidence intervals for differences in means rely on random sampling and approximate normality for validity. The interval's exclusion of zero suggests evidence against equal means, with the confidence level representing the method's long-term capture rate of the true difference.

Question 2

1. We are 90% confident that the true difference in mean sodium intake, $\mu_A-\mu_B$, is between 120 and 340 mg/day. (correct answer)
2. There is a 90% chance that Diet A adults have sodium intake 120 to 340 mg/day higher than Diet B adults.
3. We are 90% confident that $\mu_B-\mu_A$ is between 120 and 340 mg/day.
4. Because the interval is positive, 90% of adults on Diet A consume between 120 and 340 mg/day more sodium than adults on Diet B.
5. Since 0 is not in the interval, there is a 90% probability that the computed sample difference will be between 120 and 340 mg/day in future samples.

Explanation: This question asks about interpreting a confidence interval for μ_A - μ_B. The interval (120, 340) means we're 90% confident that Diet A's mean sodium intake is between 120 and 340 mg/day higher than Diet B's. Choice A correctly states this interpretation. Choice B incorrectly assigns probability to individual adults. Choice C reverses the order of subtraction. Choice D misinterprets the interval as describing individual differences. Choice E incorrectly relates the interval to future sample statistics. Key concept: confidence intervals estimate population parameters, not individual values or sample statistics.

Question 3

1. We are 92% confident that $\mu_D-\mu_T$ is between 3 and 11 minutes.
2. We are 92% confident that public-transit commuters have a population mean commute time between 3 and 11 minutes longer than drivers. (correct answer)
3. There is a 92% chance that the true difference in means is exactly 7 minutes, the midpoint of the interval.
4. About 92% of all transit commuters take between 3 and 11 minutes longer than all drivers.
5. Since 0 is not in the interval, at least 92% of individual transit commute times exceed individual driving commute times.

Explanation: In AP Statistics, this problem involves a 92% confidence interval for μ_T - μ_D from 3 to 11 minutes. Choice B correctly interprets we are 92% confident transit commuters average 3 to 11 minutes longer than drivers in the population. The positive endpoints indicate transit likely takes longer on average. Choice D is a distractor, confusing the mean difference with comparisons of all individuals. Mini-lesson: Confidence intervals for differences in means rely on random sampling and approximate normality for validity. The interval's exclusion of zero suggests evidence against equal means, with the confidence level representing the method's long-term capture rate of the true difference.

Question 4

1. There is a 90% probability that Plan P improves each athlete's time by between 0.02 and 0.15 seconds more than Plan Q.
2. We are 90% confident that the population mean improvement under Plan P is between 0.02 and 0.15 seconds.
3. We are 90% confident that, in the population, the mean improvement for Plan P exceeds that for Plan Q by between 0.02 and 0.15 seconds. (correct answer)
4. Because 0 is not in the interval, there is a 90% chance that $\mu_P=\mu_Q$.
5. We are 90% confident that $\mu_Q-\mu_P$ is between 0.02 and 0.15 seconds.

Explanation: AP Statistics here focuses on interpreting a 90% confidence interval for μ_P - μ_Q from 0.02 to 0.15 seconds. Choice C accurately conveys 90% confidence that Plan P's mean improvement exceeds Plan Q's by 0.02 to 0.15 seconds in the population. The positive endpoints exclude zero, suggesting a difference. Choice D is a distractor, wrongly implying a chance of equality despite the interval excluding zero. Mini-lesson: For two independent groups, the CI estimates the difference in population means with a range reflecting sampling error. Excluding zero aligns with rejecting the null of no difference at alpha = 0.10, emphasizing CIs as tools for inference on means rather than individual outcomes.

Question 5

1. There is a 92% probability that Treatment A reduces the recovery time by between 0.8 and 3.4 days for any given patient.
2. We are 92% confident that the true difference in population mean recovery time $(\mu_{A}-\mu_{B})$ is between $-3.4$ and $-0.8$ days, so Treatment A has a shorter mean recovery time than Treatment B. (correct answer)
3. We are 92% confident that Treatment B has a shorter mean recovery time than Treatment A by between 0.8 and 3.4 days.
4. Because the interval is entirely negative, 0 is contained in the interval.
5. We are 92% confident that the mean recovery time for Treatment A is between $-3.4$ and $-0.8$ days.

Explanation: This question involves interpreting the interval (-3.4, -0.8) for μ_A - μ_B. Since both endpoints are negative, Treatment A has a shorter mean recovery time than Treatment B. Choice B correctly interprets this negative interval as Treatment A having 0.8 to 3.4 days shorter recovery time. Choice A uses incorrect probability language about individual patients, Choice C gets the direction wrong (B doesn't have shorter recovery time), Choice D makes a nonsensical statement about 0 being in an entirely negative interval, and Choice E incorrectly interprets negative days for a single treatment's mean.

Question 6

1. There is a 90% probability that Plan P improves each athlete's time by between 0.02 and 0.15 seconds more than Plan Q.
2. We are 90% confident that the population mean improvement under Plan P is between 0.02 and 0.15 seconds.
3. We are 90% confident that, in the population, the mean improvement for Plan P exceeds that for Plan Q by between 0.02 and 0.15 seconds. (correct answer)
4. Because 0 is not in the interval, there is a 90% chance that $\mu_P=\mu_Q$.
5. We are 90% confident that $\mu_Q-\mu_P$ is between 0.02 and 0.15 seconds.

Explanation: AP Statistics here focuses on interpreting a 90% confidence interval for μ_P - μ_Q from 0.02 to 0.15 seconds. Choice C accurately conveys 90% confidence that Plan P's mean improvement exceeds Plan Q's by 0.02 to 0.15 seconds in the population. The positive endpoints exclude zero, suggesting a difference. Choice D is a distractor, wrongly implying a chance of equality despite the interval excluding zero. Mini-lesson: For two independent groups, the CI estimates the difference in population means with a range reflecting sampling error. Excluding zero aligns with rejecting the null of no difference at alpha = 0.10, emphasizing CIs as tools for inference on means rather than individual outcomes.

Question 7

1. We are 95% confident that Variety 2 has a mean yield 4.5 to 9.0 bushels per acre higher than Variety 1.
2. There is a 95% probability that the true mean difference in yield is between 4.5 and 9.0 bushels per acre.
3. We are 95% confident that the true difference in mean yield, $\mu_1-\mu_2$, is between 4.5 and 9.0 bushels per acre. (correct answer)
4. Because 0 is not in the interval, 95% of fields planted with Variety 1 will out-yield Variety 2 by 4.5 to 9.0 bushels per acre.
5. In 95% of future samples, the interval will be exactly $(4.5,\,9.0)$.

Explanation: This question asks about interpreting a positive interval (4.5, 9.0) for μ_1 - μ_2. Choice C correctly states we're 95% confident the true difference in mean yield is between 4.5 and 9.0 bushels per acre. Choice A reverses which variety has higher yield. Choice B incorrectly assigns probability to the parameter. Choice D misinterprets the interval as applying to individual fields. Choice E incorrectly claims future intervals will be identical. Remember: confidence intervals vary from sample to sample, but we're confident about the true parameter value.

Question 8

1. We are 95% confident that the true difference in mean battery life, $\mu_P-\mu_Q$, is between 0.0 and 1.8 hours. (correct answer)
2. There is a 95% probability that $\mu_P-\mu_Q$ is exactly 0.9 hours.
3. Because 0.0 is an endpoint, we are 95% confident there is definitely no difference in mean battery life between P and Q.
4. We are 95% confident that $\mu_Q-\mu_P$ is between 0.0 and 1.8 hours.
5. About 95% of all phones running P have battery life between 0.0 and 1.8 hours longer than phones running Q.

Explanation: This question tests interpretation when 0 is an endpoint: (0.0, 1.8) for μ_P - μ_Q. Choice A correctly states we're 95% confident the true difference in mean battery life is between 0.0 and 1.8 hours. Choice B incorrectly assigns probability to a specific value. Choice C wrongly concludes no difference from 0 being an endpoint; the interval suggests P likely has longer battery life. Choice D reverses the subtraction order. Choice E misapplies to individual phones. Remember: when 0 is an endpoint, we're on the borderline of concluding a directional difference.

Question 9

1. We are 98% confident that caffeine increases mean reaction time by between 5 and 30 ms compared with decaf.
2. We are 98% confident that caffeine decreases mean reaction time by between 5 and 30 ms compared with decaf. (correct answer)
3. There is a 98% chance that the true mean reaction time for caffeine is between -30 and -5 ms.
4. Because the interval does not include 0, 98% of individuals react between 5 and 30 ms faster after caffeine than after decaf.
5. If the experiment were repeated, 98% of all participants would show reaction times between -30 and -5 ms.

Explanation: This question involves interpreting a negative confidence interval for reaction times. The interval (-30, -5) for (μ_caff - μ_decaf) is entirely negative, meaning caffeine produces lower mean reaction times. Since lower reaction time means faster reactions, Choice B correctly states that caffeine decreases mean reaction time by between 5 and 30 ms. Choice A incorrectly interprets negative values as increased (slower) reaction time. Choice D wrongly applies the interval to individual participants. Choice E misunderstands what the interval represents. In reaction time studies, remember that lower values indicate better (faster) performance, so negative differences favor the first group.

Question 10

1. We are 99% confident that wells near farms have a mean nitrate concentration between 0.05 and 0.40 mg/L higher than wells far from farms. (correct answer)
2. There is a 99% probability that the sample mean difference is between 0.05 and 0.40 mg/L.
3. Because the interval is positive, 99% of near-farm wells have nitrate levels 0.05 to 0.40 mg/L higher than far-farm wells.
4. We are 99% confident that $(\mu_{\text{far}}-\mu_{\text{near}})$ is between 0.05 and 0.40 mg/L.
5. If the study were repeated, 99% of the time the true mean difference would change to fall between 0.05 and 0.40 mg/L.

Explanation: This question tests interpretation of a confidence interval comparing nitrate levels near and far from farms. The interval (0.05, 0.40) for (μ_near - μ_far) is entirely positive, indicating wells near farms have higher mean nitrate concentration. Choice A correctly states we're 99% confident that near-farm wells have mean nitrate levels between 0.05 and 0.40 mg/L higher than far-farm wells. Choice C makes the error of applying the interval to individual wells rather than population means. Choice D reverses the order of subtraction, which would make the interval negative. Choice E misunderstands confidence intervals by suggesting the true parameter changes. Remember: the true difference is fixed; it's our interval that varies with repeated sampling.

Question 11

1. We are 95% confident that Tool X takes between 1.5 and 6.0 minutes longer on average than Tool Y.
2. We are 95% confident that Tool X takes between 1.5 and 6.0 minutes less on average than Tool Y. (correct answer)
3. There is a 95% chance that $\mu_X-\mu_Y$ is negative, since the interval is negative.
4. Because 0 is not in the interval, exactly 95% of all assembly times using Tool X are between 1.5 and 6.0 minutes less than assembly times using Tool Y.
5. We are 95% confident that the difference in sample means $(\bar{x}_X-\bar{x}_Y)$ is between -6.0 and -1.5 minutes for any future samples.

Explanation: This question involves interpreting a negative confidence interval for assembly times. The interval (-6.0, -1.5) for (μ_X - μ_Y) is entirely negative, meaning Tool X has a lower mean assembly time than Tool Y. Since lower time means faster assembly, Choice B correctly states that Tool X takes between 1.5 and 6.0 minutes less on average than Tool Y. Choice A incorrectly interprets the negative values as Tool X taking longer. Choice D wrongly applies the interval to individual assembly times rather than means. Choice E misunderstands confidence intervals by suggesting they predict future sample statistics. When interpreting negative intervals, carefully consider what the negative sign means in context—here, negative means less time, which is better.

Question 12

1. We are 92% confident that the true mean battery life difference $(\mu_X-\mu_Y)$ is between -1.0 and 3.4 hours. (correct answer)
2. There is a 92% probability that Model X lasts between 1.0 and 3.4 hours longer than Model Y.
3. Because 0 is in the interval, we can conclude Model X and Model Y have exactly the same mean battery life.
4. We are 92% confident that the true mean difference $(\mu_Y-\mu_X)$ is between -1.0 and 3.4 hours.
5. 92% of all phones have battery lives that differ by between -1.0 and 3.4 hours depending on the model.

Explanation: This question tests interpretation of a confidence interval that spans both negative and positive values. The interval (-1.0, 3.4) for μ_X - μ_Y includes zero, meaning we cannot determine which battery model has a longer mean life at the 92% confidence level. Choice A correctly interprets the interval with appropriate confidence language and parameter notation. Choice C incorrectly concludes the means are exactly equal just because zero is in the interval; we can only say we lack evidence of a difference. Choice B uses incorrect probability language about individual phones. When zero falls within a confidence interval for a mean difference, we cannot reject the null hypothesis of equal population means at the corresponding significance level (in this case, α = 0.08).

Question 13

1. We are 90% confident that the mean commute time for transit riders is between 5 and 18 minutes.
2. There is a 90% probability that the true mean difference $(\mu_{\text{transit}}-\mu_{\text{drive}})$ is between 5 and 18 minutes.
3. We are 90% confident that the true mean difference in commute time $(\mu_{\text{transit}}-\mu_{\text{drive}})$ is between 5 and 18 minutes, suggesting transit has a longer mean commute time than driving. (correct answer)
4. Because 0 is not in the interval, transit riders and drivers have the same mean commute time.
5. 90% of individual transit riders have commutes that are 5 to 18 minutes longer than individual drivers.

Explanation: This question tests interpretation of a confidence interval with contextual understanding. The interval (5, 18) for μ_transit - μ_drive contains only positive values, indicating transit users have longer mean commute times. Choice C provides the complete correct interpretation: confidence language, proper parameter notation, and the contextual conclusion about transit having longer commutes. Choice B incorrectly uses probability language about the parameter. Choice E misapplies the interval to individual commuters rather than population means. When a confidence interval for μ₁ - μ₂ contains only positive values, we can conclude at the given confidence level that the first population has a larger mean than the second population.

Question 14

1. We are 99% confident that Program 1 reduces mean completion time by between 2.0 and 8.5 minutes compared with Program 2.
2. There is a 99% probability that Program 1's mean completion time is between 2.0 and 8.5 minutes less than Program 2's mean.
3. Because the interval is negative, Program 1 has a larger mean completion time than Program 2 by 2.0 to 8.5 minutes.
4. We are 99% confident that the true difference $\mu_{\text{Program 1}}-\mu_{\text{Program 2}}$ is between $-8.5$ and $-2.0$ minutes. (correct answer)
5. About 99% of employees will complete the task between 2.0 and 8.5 minutes faster after Program 1 than after Program 2.

Explanation: This question involves interpreting a negative confidence interval for μ_Program1 - μ_Program2. The interval (-8.5, -2.0) being entirely negative means Program 1 has a lower mean completion time than Program 2. Choice D correctly states we're 99% confident the true difference is between -8.5 and -2.0 minutes. Choice A misinterprets the direction. Choice B incorrectly assigns probability to the parameter. Choice C gets the direction wrong. Choice E incorrectly applies the interval to individuals. Remember: negative values for μ_1 - μ_2 mean μ_1 < μ_2.

Question 15

1. We are 95% confident that patients on the new therapy recover, on average, between 0.6 and 2.4 days sooner than patients on standard therapy, in the population. (correct answer)
2. There is a 95% probability that the new therapy reduces every patient's recovery time by between 0.6 and 2.4 days.
3. We are 95% confident that $\mu_S-\mu_N$ is between $-2.4$ and $-0.6$ days.
4. Because the interval is negative, the standard therapy has a smaller mean recovery time than the new therapy.
5. There is a 95% chance that $\mu_N-\mu_S$ will fall between $-2.4$ and $-0.6$ in a future sample.

Explanation: This AP Statistics question evaluates a 95% confidence interval for μ_N - μ_S from -2.4 to -0.6 days. Choice A correctly states we are 95% confident new therapy patients recover 0.6 to 2.4 days sooner on average. The negative endpoints imply shorter mean recovery for the new therapy. A distractor is choice D, which misinterprets the negative interval as standard therapy being faster. Mini-lesson: CIs for μ1 - μ2 are built from sample means and variances, using t-distributions for critical values. A fully negative interval supports μ1 < μ2, but interpretations must avoid causal claims unless from experiments, focusing instead on population parameter estimates.

Question 16

1. We are 90% confident that Layout A's mean time is between 1.0 and 6.5 seconds less than Layout B's mean time. (correct answer)
2. There is a 90% chance that $\mu_A-\mu_B$ is between $-6.5$ and $-1.0$ seconds.
3. Because the interval is negative, Layout A must be slower on average than Layout B.
4. We are 90% confident that the difference in sample means, $\bar{x}_A-\bar{x}_B$, is between $-6.5$ and $-1.0$ seconds.
5. Since 0 is not in the interval, there is no overlap in individual user times between the two layouts.

Explanation: This AP Statistics question focuses on interpreting a confidence interval for the difference in means, μ_A - μ_B, with endpoints -6.5 and -1.0 seconds. The negative values indicate Layout A likely has a lower mean time, and choice A correctly rephrases this as A being 1.0 to 6.5 seconds faster. We are 90% confident that the true population difference lies between these endpoints. Choice B is a distractor because it incorrectly assigns a 'chance' to the fixed interval containing the parameter, confusing probability with confidence. Mini-lesson: Confidence intervals for μ1 - μ2 are constructed from independent samples, using the formula (x̄1 - x̄2) ± t* √(s1²/n1 + s2²/n2), assuming normality or large samples. The sign of the interval informs which mean is larger, and excluding zero supports a difference at the (1 - confidence level) significance.

Question 17

1. We are 95% confident that employees who work remotely 2 days/week have a population mean commute time between 1 and 6 minutes less than employees who never work remotely. (correct answer)
2. There is a 95% probability that $(\mu_{remote}-\mu_{never})$ is between $-6$ and $-1$ minutes for the next sample taken.
3. Because the entire interval is negative, the sample means must have been equal.
4. We are 95% confident that $(\mu_{never}-\mu_{remote})$ is between $-6$ and $-1$ minutes.
5. Since 0 is not included, there is no statistically significant difference in the population means.

Explanation: This question involves interpreting a negative confidence interval for the difference in mean commute times. The interval (-6, -1) for (μ_remote - μ_never) is entirely negative, indicating μ_remote < μ_never. This means employees who work remotely have a shorter average commute time. The correct answer states we are 95% confident that employees who work remotely 2 days/week have a population mean commute time between 1 and 6 minutes less than those who never work remotely. Choice E incorrectly interprets the exclusion of 0 as meaning there's no significant difference—actually, excluding 0 indicates there IS a significant difference. Remember to pay attention to the order of subtraction in the difference: (μ_remote - μ_never) being negative means remote workers have shorter commutes.

Question 18

1. We are 95% confident that homes with smart thermostats use, on average, between 30 kWh less and 10 kWh more per month than homes without smart thermostats. (correct answer)
2. Because 0 is included, we are 95% confident that smart thermostats increase electricity use.
3. There is a 95% chance that the true difference in means is between $-30$ and $10$ for the homes in the samples only, not for the populations.
4. We are 95% confident that $(\mu_{no\ smart}-\mu_{smart})$ is between $-30$ and $10$ kWh.
5. 95% of individual homes with smart thermostats differ from individual homes without by between $-30$ and $10$ kWh.

Explanation: This question tests interpretation of a confidence interval containing zero. The interval (-30, 10) for (μ_smart - μ_no smart) includes both negative and positive values. This means smart thermostats could either decrease electricity use (by up to 30 kWh) or increase it (by up to 10 kWh). The correct interpretation states we are 95% confident that homes with smart thermostats use, on average, between 30 kWh less and 10 kWh more per month than homes without. Choice B incorrectly concludes that including 0 means smart thermostats increase use—we cannot determine the direction of the effect. When a confidence interval spans zero, it indicates uncertainty about whether the first group's mean is higher or lower than the second group's mean.

Question 19

1. There is a 95% probability that the true mean difference $(\mu_{\text{app}}-\mu_{\text{no app}})$ is between 1.2 and 3.8 hours.
2. We are 95% confident that the mean weekly study time for app users is between 1.2 and 3.8 hours more than for non-users. (correct answer)
3. We are 95% confident that the mean weekly study time for non-users is between 1.2 and 3.8 hours more than for app users.
4. Because 0 is not in the interval, 95% of individual students who use the app study between 1.2 and 3.8 hours more than individual non-users.
5. If we take many samples, 95% of the intervals we compute will have endpoints 1.2 and 3.8 hours.

Explanation: This question tests understanding of confidence intervals for the difference of two means. The interval (1.2, 3.8) was calculated for (μ_app - μ_no app), meaning we're subtracting the non-app users' mean from the app users' mean. Since both endpoints are positive, this indicates the app users have a higher mean study time. Choice B correctly states we're 95% confident that app users study between 1.2 and 3.8 hours more per week than non-users. Choice A incorrectly uses probability language instead of confidence, while Choice D wrongly interprets the interval as applying to individual students rather than population means. Remember: confidence intervals estimate population parameters (means), not individual values, and we express our conclusion using confidence language, not probability.

Question 20

1. We are 90% confident that the true difference in population means $(\mu_A-\mu_B)$ is between -4.5 and 1.0 points. (correct answer)
2. There is a 90% chance that $(\mu_A-\mu_B)$ is between -4.5 and 1.0 points for these two populations.
3. Because the interval includes 0, Method A is proven to be equal to Method B in the populations.
4. We are 90% confident that Method A increases each individual student's score by between -4.5 and 1.0 points compared with Method B.
5. We are 90% confident that $(\mu_B-\mu_A)$ is between -4.5 and 1.0 points.

Explanation: This question involves interpreting a confidence interval for the difference between two teaching methods. The interval (-4.5, 1.0) for (μ_A - μ_B) contains both negative and positive values, including zero. Choice A correctly interprets this as being 90% confident that the true difference in population means is between -4.5 and 1.0 points. Choice B incorrectly uses probability language, while Choice C makes the common error of claiming equality when zero is in the interval—we can only say there's no significant difference, not that they're equal. Choice E reverses the order of subtraction, which would change the signs. When an interval contains zero, it means we cannot determine which method is better based on this data alone.