Confidence Intervals: Difference of Two Means - AP Statistics

A company compares mean battery life (hours) of phones with Battery Model X versus Model Y. Using independent random samples, a 92% confidence interval for $(\mu_X-\mu_Y)$ is $(-1.0,\ 3.4)$ . Which interpretation is correct?
Answer: We are 92% confident that the true mean battery life difference $(\mu_X-\mu_Y)$ is between -1.0 and 3.4 hours. 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).
A fitness study compares mean resting heart rate (bpm) for people who do yoga weekly vs people who do not. From two independent random samples, a 98% confidence interval for $(\mu_{yoga}-\mu_{no\ yoga})$ is $(-4.5, 0.8)$ . Which interpretation is correct?
Answer: We are 98% confident that the population mean resting heart rate for weekly yoga is between 4.5 bpm lower and 0.8 bpm higher than for no yoga. This question tests understanding of a confidence interval that contains zero. The interval (-4.5, 0.8) for (μ_yoga - μ_no yoga) includes both negative and positive values, spanning zero. This means we cannot determine whether yoga practitioners have higher or lower average resting heart rates at the 98% confidence level. The correct answer states we are 98% confident that the population mean resting heart rate for weekly yoga is between 4.5 bpm lower and 0.8 bpm higher than for no yoga. Choice B incorrectly concludes that including 0 means yoga has no effect on any individual—the interval describes population means, not individual effects. When zero is included in a confidence interval for a difference, it indicates we cannot reject the null hypothesis of no difference at the given confidence level.
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?
Answer: 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. 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 researcher compared mean reaction time (milliseconds) for people who drank caffeinated coffee versus decaf. From independent random samples, a 90% confidence interval for $(\mu_{\text{caff}}-\mu_{\text{decaf}})$ was $( -35,\ -5)$ . Which interpretation is correct?
Answer: We are 90% confident that the true difference in population mean reaction times $(\mu_{\text{caff}}-\mu_{\text{decaf}})$ is between $-35$ and $-5$ ms, so the caffeinated group has a smaller (faster) mean reaction time. This question asks about interpreting the interval (-35, -5) for μ_caff - μ_decaf. Since both endpoints are negative, the caffeinated group has faster (smaller) mean reaction times. Choice B correctly interprets this as the caffeinated group having 5 to 35 ms faster reactions. Choice A gets the direction wrong (caffeine makes reactions faster, not slower), Choice C incorrectly assigns probability to a specific value, Choice D misinterprets negative milliseconds for a single group's mean, and Choice E misinterprets what 0 not being in the interval means (it indicates there IS a difference).
To compare two fertilizers, an agronomist measured the mean height of plants after 6 weeks for independent plots using Fertilizer X and Fertilizer Y. A 95% confidence interval for $\mu_X-\mu_Y$ is $( -1.5,\ 0.9)$ cm. Which interpretation is correct?
Answer: Because 0 is in the interval, there is not convincing evidence that the population mean heights differ between Fertilizer X and Fertilizer Y. This AP Statistics question tests confidence interval interpretation for μ_X - μ_Y, with a 95% interval from -1.5 to 0.9 cm including zero. Choice A correctly notes this implies no convincing evidence of a mean height difference. The endpoints span negative to positive values, consistent with possible equality. A distractor is choice C, which misstates the interval as positive and reverses the comparison. Mini-lesson: For independent samples, the CI formula incorporates standard errors from each group, often using unpooled t-methods if variances differ. Including zero means the data do not support a significant difference, reinforcing that CIs quantify uncertainty in estimating population parameters.
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?
Answer: 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. 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 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?
Answer: We are 90% confident that the true difference in mean sodium intake, $\mu_A-\mu_B$ , is between 120 and 340 mg/day. 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 nutritionist compares mean daily sodium intake (mg) for adults who follow Diet A versus Diet B. From independent random samples, a 99% confidence interval for $(\mu_{A}-\mu_{B})$ is $(-420,\ -110)$ . Which interpretation is correct?
Answer: We are 99% confident that the true mean difference $(\mu_{A}-\mu_{B})$ is between -420 and -110 mg, meaning Diet A has a lower mean sodium intake than Diet B. This question tests understanding of negative confidence intervals for mean differences. The interval (-420, -110) for μ_A - μ_B contains only negative values, indicating that Diet A has a lower mean sodium intake than Diet B. Choice C correctly interprets both the confidence level and the direction of the difference. Choice A reverses the interpretation by stating Diet A has higher sodium intake. Choice E incorrectly applies the interval to individual adults rather than population means. When interpreting confidence intervals with all negative values, remember that if μ₁ - μ₂ is negative, then μ₁ < μ₂. In this case, adults on Diet A consume between 110 and 420 mg less sodium per day on average compared to Diet B.
A hospital compares mean length of stay (days) for patients receiving a new discharge protocol versus the standard protocol. From independent random samples, a 90% confidence interval for $(\mu_{\text{new}}-\mu_{\text{standard}})$ is $(-2.4,\ -0.6)$ . Which interpretation is correct?
Answer: We are 90% confident that the true mean difference in length of stay $(\mu_{\text{new}}-\mu_{\text{standard}})$ is between -2.4 and -0.6 days, suggesting the new protocol reduces mean length of stay. This question assesses interpretation of negative confidence intervals in a medical context. The interval (-2.4, -0.6) for μ_new - μ_standard contains only negative values, indicating the new protocol reduces mean length of stay (since negative differences mean the new protocol has shorter stays). Choice B correctly interprets the interval with confidence language and explains the practical meaning. Choice D incorrectly reverses the parameter order, which would change the sign of the interval. Choice E misinterprets the evidence by claiming the standard protocol is better. When interpreting confidence intervals in contexts where smaller values are desirable (like hospital stays), negative values for μ_treatment - μ_control indicate the treatment is beneficial.
Two independent random samples are taken to compare mean commute time (minutes) for employees who work remotely 2 days/week vs employees who never work remotely. A 95% confidence interval for $(\mu_{remote}-\mu_{never})$ is $( -6, -1 )$ . Which interpretation is correct?
Answer: 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. 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.
A city compared mean commute times (minutes) for residents who take public transit versus those who drive. From independent random samples, a 95% confidence interval for $(\mu_{\text{transit}}-\mu_{\text{drive}})$ was $( -6.5,\ 2.0)$ . Which interpretation is correct?
Answer: We are 95% confident that the true difference in population mean commute times $(\mu_{\text{transit}}-\mu_{\text{drive}})$ is between $-6.5$ and $2.0$ minutes. This question tests understanding of a confidence interval for μ_transit - μ_drive = (-6.5, 2.0). The interval includes 0, meaning transit could have either shorter or longer mean commute times than driving. Choice C correctly interprets this as our confidence about the true population mean difference. Choice A only mentions one group's mean, Choice B makes too strong a claim about no difference existing, Choice D incorrectly assigns probability to a specific parameter value, and Choice E misinterprets the interval (it doesn't say drivers commute longer, and the values are wrong).
A clinic compares mean systolic blood pressure (mmHg) for patients taking Drug A versus Drug B. From independent random samples, a 92% confidence interval for $(\mu_A-\mu_B)$ is $(-7,\,1)$ . Which interpretation is correct?
Answer: We are 92% confident that the true difference in mean systolic blood pressure, $\mu_A-\mu_B$ , is between $-7$ and $1$ mmHg. This question involves a mostly negative interval (-7, 1) for μ_A - μ_B. Choice D correctly states we're 92% confident the true difference in mean systolic blood pressure is between -7 and 1 mmHg. Choice A misinterprets the negative values. Choice B incorrectly relates confidence level to probability about the parameter. Choice C wrongly concludes no effect from 0 being in the interval. Choice E misapplies the interval to individuals. Key concept: intervals containing 0 mean we're not confident about the direction of difference, not that there's no difference.
A researcher compared mean reaction times (milliseconds) for two independent groups: participants who drank coffee (C) and participants who drank water (W). A 90% confidence interval for $\mu_C-\mu_W$ is $( -12,\ 30)$ . Which interpretation is correct?
Answer: We are 90% confident that the difference in population means, $\mu_C-\mu_W$ , is between $-12$ and 30 milliseconds. This question in AP Statistics assesses confidence interval interpretation for μ_C - μ_W, with a 90% interval from -12 to 30 milliseconds. Choice B correctly conveys 90% confidence that the true difference falls between these endpoints, which straddle zero. The endpoints capture a range including possible negative, zero, or positive differences. A distractor is choice D, which overstates that including zero means exactly equal means, ignoring the interval's width. Mini-lesson: Constructing CIs for mean differences involves pooling or separate variances, depending on assumptions, and uses t-critical values for small samples. Intervals including zero do not support rejecting equality, highlighting the link between CIs and hypothesis testing for differences.
A nutrition researcher compared mean sodium intake (mg/day) for adults following Diet X versus Diet Y. From independent random samples, a 98% confidence interval for $(\mu_{X}-\mu_{Y})$ was $( -220,\ -40)$ . Which interpretation is correct?
Answer: We are 98% confident that the true difference in population means $(\mu_{X}-\mu_{Y})$ is between $-220$ and $-40$ mg/day, so Diet X has a lower mean sodium intake than Diet Y. This question involves interpreting a confidence interval for μ_X - μ_Y = (-220, -40). Since both endpoints are negative, Diet X has lower mean sodium intake than Diet Y by 40 to 220 mg/day. Choice B correctly interprets this negative interval. Choice A incorrectly states Diet X has higher sodium (the opposite of what the negative interval shows), Choice C misuses probability language about the sample mean, Choice D misinterprets the absence of 0 (which actually indicates a significant difference exists), and Choice E reverses the subtraction order which would make the interval positive.
A school district compares mean math test scores for students taught with Method A versus Method B. From two independent random samples, a 90% confidence interval for $(\mu_A-\mu_B)$ is $(-4.5,\ 1.0)$ points. Which interpretation is correct?
Answer: We are 90% confident that the true difference in population means $(\mu_A-\mu_B)$ is between -4.5 and 1.0 points. 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.
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?
Answer: We are 98% confident that caffeine decreases mean reaction time by between 5 and 30 ms compared with decaf. 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.
A psychologist compares mean reaction time (milliseconds) after drinking caffeinated tea versus decaffeinated tea. From independent random samples, a 98% confidence interval for $(\mu_{\text{caff}}-\mu_{\text{decaf}})$ is $(-25,\ -5)$ . Which interpretation is correct?
Answer: We are 98% confident that the true mean difference in reaction time $(\mu_{\text{caff}}-\mu_{\text{decaf}})$ is between -25 and -5 ms, meaning caffeinated tea is associated with a smaller mean reaction time. This question examines interpretation of negative confidence intervals in a reaction time context. The interval (-25, -5) for μ_caff - μ_decaf contains only negative values, and since smaller reaction times are better, this indicates caffeinated tea improves (reduces) mean reaction time. Choice A correctly interprets the interval with proper confidence language and explains that negative values mean smaller (better) reaction times for the caffeinated group. Choice D incorrectly reverses the interpretation. Choice B wrongly applies the interval to every individual rather than population means. In reaction time studies, remember that smaller values indicate faster responses, so negative differences favor the first group when measuring μ₁ - μ₂.
A school district compares mean time (minutes) to complete a standardized math test for students who take the test on a computer versus on paper. From independent random samples, a 95% confidence interval for $(\mu_{\text{computer}}-\mu_{\text{paper}})$ is $(0.5,\ 6.0)$ . Which interpretation is correct?
Answer: We are 95% confident that the computer format increases the mean completion time by between 0.5 and 6.0 minutes compared with paper. This question examines interpretation of a positive confidence interval for mean differences. The interval (0.5, 6.0) for μ_computer - μ_paper contains only positive values, indicating computer testing takes longer on average. Choice A correctly states that we are 95% confident the computer format increases mean completion time by the given range. Choice B incorrectly uses probability language for individual students rather than confidence language for the population parameter. Choice D contradicts the evidence by claiming no difference when zero is not in the interval. When all values in a confidence interval are positive, we have evidence that the first group's mean is larger than the second group's mean at the specified confidence level.
A coach compares mean vertical jump height (cm) for athletes following Training Program P versus Program Q. From independent random samples, a 95% confidence interval for $(\mu_P-\mu_Q)$ is $(2.0,\ 9.5)$ . Which interpretation is correct?
Answer: We are 95% confident that the true mean difference in vertical jump height $(\mu_P-\mu_Q)$ is between 2.0 and 9.5 cm. This question assesses understanding of positive confidence intervals and proper interpretation language. The interval (2.0, 9.5) for μ_P - μ_Q contains only positive values, indicating Program P produces higher mean vertical jump heights. Choice B correctly uses confidence language about the true mean difference between populations. Choice A incorrectly interprets the interval as applying to individual athletes rather than population means. Choice D reverses the direction, claiming Program Q is better when the positive interval actually favors Program P. When interpreting confidence intervals, always use "confident" language about population parameters, not "probability" language about individual observations or sample statistics.
A company tests two website designs. Using independent random samples of users, they construct a 90% confidence interval for the difference in mean time on site (in minutes), defined as $(\mu_{A}-\mu_{B})$ , and obtain $(0.5, 2.1)$ . Which interpretation is correct?
Answer: We are 90% confident that Design A increases the population mean time on site by between $0.5$ and $2.1$ minutes compared with Design B. This question asks about interpreting a confidence interval for the difference in mean time on site between two website designs. The interval (0.5, 2.1) for (μ_A - μ_B) is entirely positive, meaning μ_A > μ_B. This indicates Design A has a higher population mean time on site than Design B. The correct answer states we are 90% confident that Design A increases the population mean time on site by between 0.5 and 2.1 minutes compared to Design B. Choice C incorrectly applies the interval to individual users rather than population means. Remember that confidence intervals describe our uncertainty about population parameters, not the distribution of individual observations or the probability that the parameter equals a specific value.

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