Confidence Intervals: Difference of Two Means
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AP Statistics › Confidence Intervals: Difference of Two Means
A city compared mean commute times for two independent groups: commuters who use public transit (T) and commuters who drive (D). A 92% confidence interval for $\mu_T-\mu_D$ is $(3,\ 11)$ minutes. Which interpretation is correct?
There is a 92% chance that the true difference in means is exactly 7 minutes, the midpoint of the interval.
We are 92% confident that $\mu_D-\mu_T$ is between 3 and 11 minutes.
About 92% of all transit commuters take between 3 and 11 minutes longer than all drivers.
Since 0 is not in the interval, at least 92% of individual transit commute times exceed individual driving commute times.
We are 92% confident that public-transit commuters have a population mean commute time between 3 and 11 minutes longer than drivers.
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.
A nutritionist compares mean sodium intake (mg/day) for adults following Diet A versus Diet B. Using independent random samples, a 90% confidence interval for the difference in population means was found to be $(\mu_A-\mu_B)\in(120,,340)$. Which interpretation is correct?
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.
We are 90% confident that $\mu_B-\mu_A$ is between 120 and 340 mg/day.
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.
There is a 90% chance that Diet A adults have sodium intake 120 to 340 mg/day higher than Diet B adults.
We are 90% confident that the true difference in mean sodium intake, $\mu_A-\mu_B$, is between 120 and 340 mg/day.
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.
A city compared mean commute times for two independent groups: commuters who use public transit (T) and commuters who drive (D). A 92% confidence interval for $\mu_T-\mu_D$ is $(3,\ 11)$ minutes. Which interpretation is correct?
Since 0 is not in the interval, at least 92% of individual transit commute times exceed individual driving commute times.
We are 92% confident that public-transit commuters have a population mean commute time between 3 and 11 minutes longer than drivers.
About 92% of all transit commuters take between 3 and 11 minutes longer than all drivers.
There is a 92% chance that the true difference in means is exactly 7 minutes, the midpoint of the interval.
We are 92% confident that $\mu_D-\mu_T$ is between 3 and 11 minutes.
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.
A coach compared mean improvement (seconds) in a 100-meter sprint after two different training plans, using independent groups of athletes: Plan P and Plan Q. A 90% confidence interval for $\mu_P-\mu_Q$ is $(0.02,\ 0.15)$ seconds. Which interpretation is correct?
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.
We are 90% confident that the population mean improvement under Plan P is between 0.02 and 0.15 seconds.
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.
Because 0 is not in the interval, there is a 90% chance that $\mu_P=\mu_Q$.
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.
A hospital compared mean recovery time (days) for patients receiving Treatment A versus Treatment B. From independent random samples, a 92% confidence interval for $(\mu_{A}-\mu_{B})$ was $( -3.4,\ -0.8)$. Which interpretation is correct?
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.
We are 92% confident that the mean recovery time for Treatment A is between $-3.4$ and $-0.8$ days.
We are 92% confident that Treatment B has a shorter mean recovery time than Treatment A by between 0.8 and 3.4 days.
Because the interval is entirely negative, 0 is contained in the interval.
There is a 92% probability that Treatment A reduces the recovery time by between 0.8 and 3.4 days for any given patient.
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.
A coach compared mean improvement (seconds) in a 100-meter sprint after two different training plans, using independent groups of athletes: Plan P and Plan Q. A 90% confidence interval for $\mu_P-\mu_Q$ is $(0.02,\ 0.15)$ seconds. Which interpretation is correct?
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.
Because 0 is not in the interval, there is a 90% chance that $\mu_P=\mu_Q$.
We are 90% confident that $\mu_Q-\mu_P$ is between 0.02 and 0.15 seconds.
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.
We are 90% confident that the population mean improvement under Plan 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.
A farmer compares mean yield (bushels per acre) for Corn Variety 1 versus Variety 2. Independent random samples of fields were used to compute a 95% confidence interval for $(\mu_1-\mu_2)$ of $(4.5,,9.0)$. Which interpretation is correct?
In 95% of future samples, the interval will be exactly $(4.5,,9.0)$.
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.
There is a 95% probability that the true mean difference in yield is between 4.5 and 9.0 bushels per acre.
We are 95% confident that Variety 2 has a mean yield 4.5 to 9.0 bushels per acre higher than Variety 1.
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.
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.
A developer compares mean battery life (hours) of phones running Operating System P versus Operating System Q. Using independent random samples, a 95% confidence interval for $(\mu_P-\mu_Q)$ is $(0.0,,1.8)$. Which interpretation is correct?
About 95% of all phones running P have battery life between 0.0 and 1.8 hours longer than phones running Q.
We are 95% confident that the true difference in mean battery life, $\mu_P-\mu_Q$, is between 0.0 and 1.8 hours.
Because 0.0 is an endpoint, we are 95% confident there is definitely no difference in mean battery life between P and Q.
We are 95% confident that $\mu_Q-\mu_P$ is between 0.0 and 1.8 hours.
There is a 95% probability that $\mu_P-\mu_Q$ is exactly 0.9 hours.
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.
A psychologist compares mean reaction time (milliseconds) for participants after drinking caffeinated coffee versus decaf. Two independent random samples yield a 98% confidence interval for $(\mu_{\text{caff}}-\mu_{\text{decaf}})$ of $(-30,\ -5)$ ms. Which interpretation is correct?
Because the interval does not include 0, 98% of individuals react between 5 and 30 ms faster after caffeine than after decaf.
We are 98% confident that caffeine increases mean reaction time by between 5 and 30 ms compared with decaf.
There is a 98% chance that the true mean reaction time for caffeine is between -30 and -5 ms.
If the experiment were repeated, 98% of all participants would show reaction times between -30 and -5 ms.
We are 98% confident that caffeine decreases mean reaction time by between 5 and 30 ms compared with decaf.
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.
An environmental scientist compares mean nitrate concentration (mg/L) in water from wells near farms versus wells far from farms. Two independent random samples produce a 99% confidence interval for $(\mu_{\text{near}}-\mu_{\text{far}})$ of $(0.05,\ 0.40)$ mg/L. Which interpretation is correct?
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.
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.
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.
There is a 99% probability that the sample mean difference is between 0.05 and 0.40 mg/L.
We are 99% confident that $(\mu_{\text{far}}-\mu_{\text{near}})$ is 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.