Justifying Claims: Difference of Two Means
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AP Statistics › Justifying Claims: Difference of Two Means
A hospital compares mean length of stay (days) for patients receiving a new discharge protocol (New) versus the old protocol (Old). A 98% confidence interval for $\mu_{New}-\mu_{Old}$ is $(-1.9,\ -0.4)$. The hospital claims the new protocol reduces mean length of stay. Is the claim supported by the interval?
Yes, because 98% of patients under the new protocol stay fewer days than 98% under the old protocol.
No, because a negative interval means the new protocol increases length of stay.
Yes, because the entire interval is below 0, supporting that $\mu_{New}<\mu_{Old}$.
Yes, because the interval proves the new protocol causes every patient to leave 0.4 to 1.9 days earlier.
No, because 0 is not in the interval, so there is no difference.
Explanation
This question tests understanding of negative confidence intervals in context. The 98% confidence interval for μ_New - μ_Old is (-1.9, -0.4), which is entirely below 0. This means μ_New - μ_Old < 0, so μ_New < μ_Old, indicating that the new protocol has a lower mean length of stay. Since lower length of stay means patients leave sooner, the hospital's claim that the new protocol reduces mean length of stay is supported. Choice B incorrectly interprets negative values as increasing length of stay. Choice C misunderstands what 0 not being in the interval means. Choice D wrongly interprets the interval as describing individual patient percentiles. Choice E claims the interval applies to every patient and implies causation beyond what's justified. When μ_1 - μ_2 is entirely negative, it supports that μ_1 < μ_2.
A teacher compares mean quiz scores (percent) between a class that used practice quizzes (P) and a class that did not (N). A 95% confidence interval for $\mu_P-\mu_N$ is $(-4,\ 9)$. The teacher claims using practice quizzes increases the mean quiz score. Is the claim supported by the interval?
Yes, because the interval proves practice quizzes cause an increase between 4 and 9 points.
No, because the interval is for $\mu_N-\mu_P$ and it crosses 0.
No, because the interval includes 0, so it does not support that $\mu_P>\mu_N$.
Yes, because including 0 means there is definitely some improvement.
Yes, because the upper end of the interval is positive, so practice quizzes increase scores.
Explanation
This question examines a confidence interval that contains 0. The 95% confidence interval for μ_P - μ_N is (-4, 9), which includes 0. Since 0 is in the interval, we cannot conclude that the mean quiz scores differ between the two groups. The teacher's claim that practice quizzes increase the mean score (μ_P > μ_N) is not supported because 0 is a plausible value for the true difference. Choice A incorrectly focuses only on the positive end. Choice C misinterprets 0 being in the interval as confirming improvement. Choice D is partially correct about 0 being in the interval but incorrectly states the order. Choice E wrongly claims causation and misinterprets the interval. When 0 is contained in a confidence interval for μ_1 - μ_2, we cannot make comparative claims about the means.
A school compares mean nightly sleep for students who participate in a mindfulness program (M) versus those who do not (N). The researchers report a 95% confidence interval for the difference in population means, $\mu_M-\mu_N$, as $(0.2,\ 1.1)$ hours. A student claims: “Students in the mindfulness program sleep more on average than students not in the program.” Is the claim supported by the confidence interval?
No; because the interval contains 0, there may be no difference in mean sleep.
No; because the interval is for $\mu_N-\mu_M$, not $\mu_M-\mu_N$.
Yes; because 95% of students in the program sleep more than those not in the program.
Yes; because all values in the interval are positive, it supports $\mu_M>\mu_N$.
Yes; because the interval proves the program causes more sleep.
Explanation
This question tests understanding of how to interpret confidence intervals when comparing two population means. The 95% confidence interval for μ_M - μ_N is (0.2, 1.1) hours, where M represents students in the mindfulness program and N represents those not in the program. Since the entire interval contains only positive values, we can conclude with 95% confidence that μ_M > μ_N, meaning students in the mindfulness program sleep more on average. The key insight is that when a confidence interval for μ₁ - μ₂ contains only positive values, it supports the claim that μ₁ > μ₂. Option D incorrectly interprets the interval as describing individual students rather than population means, while option E incorrectly claims causation when the interval only shows association.
Two teaching methods are compared using mean final-exam scores. A 95% confidence interval for the difference in population means, $\mu_{\text{Flipped}}-\mu_{\text{Lecture}}$, is $( -1.5,\ 6.0 )$ points. A teacher claims: “Students in the flipped classroom score higher on average than students in the lecture classroom.” Is the claim supported by the confidence interval?
Yes; because 95% of flipped-class students score higher than lecture students.
Yes; because the interval contains 0, it confirms no difference and therefore supports the claim.
Yes; because the interval contains some values above 0, the flipped method is higher.
No; because the interval contains 0, the data do not provide convincing evidence that $\mu_{\text{Flipped}} > \mu_{\text{Lecture}}$.
No; because the interval is for $\mu_{\text{Lecture}}-\mu_{\text{Flipped}}$, not the stated order.
Explanation
This question addresses a common scenario where a confidence interval contains zero. The 95% confidence interval for μ_Flipped - μ_Lecture is (-1.5, 6.0) points, which includes negative values, zero, and positive values. Because zero is within the interval, it's plausible that μ_Flipped = μ_Lecture, meaning the data do not provide convincing evidence at the 95% confidence level that the flipped classroom produces higher mean scores. The teacher's claim requires μ_Flipped > μ_Lecture, which would need the entire interval to be positive. Option A incorrectly focuses only on the positive values while ignoring that the interval also contains negative values and zero. When interpreting confidence intervals for differences, the key question is whether zero is included—if it is, we cannot conclude there's a difference between the means.
A university compares mean GPA for students who attended supplemental instruction sessions (S) versus those who did not (N). A 90% confidence interval for $\mu_S-\mu_N$ is $(0.05,\ 0.30)$. The university claims attendees have a higher mean GPA. Is the claim supported by the interval?
Yes, because the interval proves attending causes each student’s GPA to increase by 0.05 to 0.30.
No, because 0 is not in the interval, so there is no difference.
Yes, because 90% of students who attend have higher GPAs than 90% who do not.
Yes, because the entire interval is above 0, supporting that $\mu_S>\mu_N$.
No, because the interval is small, so the difference is not real.
Explanation
This question tests interpretation of a positive confidence interval. The 90% confidence interval for μ_S - μ_N is (0.05, 0.30), which is entirely above 0. This means μ_S - μ_N > 0, so μ_S > μ_N, supporting the claim that students who attended supplemental instruction have a higher mean GPA. Choice A incorrectly dismisses the interval based on its width. Choice C misinterprets 0 not being in the interval as meaning no difference exists. Choice D wrongly interprets the interval as describing individual student percentiles. Choice E incorrectly claims causation and misinterprets the interval as applying to individual changes. When a confidence interval for μ_1 - μ_2 is entirely positive, it provides evidence that μ_1 > μ_2 at the stated confidence level.
A counselor compares mean stress-score (higher means more stress) for students in a support group (G) versus not in the group (N). A 99% confidence interval for $\mu_G-\mu_N$ is $(-5.0,\ -0.5)$. The counselor claims: “Students in the support group have lower mean stress than students not in the group.” Is the claim supported by the confidence interval?
No; because 99% confidence means there is a 99% chance the true difference is 0.
No; because the interval includes 0, so no conclusion can be made.
No; because the interval is negative, it indicates the group has higher stress.
Yes; because the interval guarantees the support group causes lower stress.
Yes; because the entire interval is negative, it supports $\mu_G<\mu_N$.
Explanation
This question involves interpreting a negative confidence interval where lower scores are better. The 99% confidence interval for μ_G - μ_N is (-5.0, -0.5), entirely negative, where G represents the support group and N represents not in the group. Since higher scores mean more stress and the interval is entirely negative, we can conclude μ_G < μ_N, meaning students in the support group have lower mean stress. The counselor's claim is supported. Option B misunderstands what confidence level means—99% confidence refers to the reliability of the interval estimation method, not the probability that the true difference is zero. Option C incorrectly interprets negative values as indicating higher stress for the group. When interpreting intervals, always consider both the direction of the difference and what constitutes a desirable outcome.
Two brands of batteries are tested for mean lifetime (hours). Brand A had a sample mean lifetime of 9.8 hours and Brand B had 10.1 hours. A 95% confidence interval for the difference in population means, defined as $\mu_A-\mu_B$, is $(-0.9, -0.1)$. A consumer claims, “Brand B lasts longer on average than Brand A.” Is the claim supported by the confidence interval?
Yes, because the entire interval is below 0, which supports $\mu_A<\mu_B$.
Yes, because 95% confidence means 95% of Brand B batteries last longer than Brand A batteries.
Yes, because the sample means differ by 0.3 hours, so Brand B must be longer in the population.
No, because the interval does not include 0, so the brands must be the same.
No, because the confidence interval is for $\mu_B-\mu_A$ rather than $\mu_A-\mu_B$.
Explanation
This question involves interpreting a negative confidence interval to support a claim about which mean is larger. The interval (-0.9, -0.1) for μ_A - μ_B is entirely negative, meaning we are 95% confident that μ_A - μ_B < 0. This is equivalent to μ_A < μ_B, which supports the consumer's claim that Brand B lasts longer on average than Brand A. Students sometimes get confused about the order of subtraction or misinterpret confidence levels as applying to individual items rather than the parameter. When comparing two means, if the confidence interval for their difference is entirely negative, it provides evidence that the first mean is less than the second mean.
A professor claims that students in a flipped classroom format have a higher mean final exam score than students in a traditional format. A 95% confidence interval for $(\mu_{\text{flipped}}-\mu_{\text{traditional}})$ is $( -3,\ 9)$ points. Is the claim supported by this interval?
Yes, because the interval includes positive values, so flipped must be higher.
No, because the interval is for $(\mu_{\text{traditional}}-\mu_{\text{flipped}})$, so it actually supports the claim.
Yes, because 95% confidence means there is a 95% chance the flipped mean is higher.
No, because the interval includes 0, so no difference is plausible.
Yes, because the interval’s midpoint is positive, so the claim is proven.
Explanation
This question asks whether flipped classrooms produce higher exam scores. The interval (-3, 9) for (μ_flipped - μ_traditional) contains 0, meaning a difference of 0 is plausible at the 95% confidence level. Since we cannot rule out the possibility that the two teaching formats produce equal mean scores, the claim that μ_flipped > μ_traditional is not supported. Choice C incorrectly focuses on the positive midpoint rather than the fact that 0 is included. When a confidence interval contains both negative and positive values, it indicates uncertainty about the direction of the difference, and no directional claim can be supported.
A pharmacist claims that a generic medication has the same mean time to symptom relief as the brand-name medication. A 95% confidence interval for $(\mu_{\text{generic}}-\mu_{\text{brand}})$ is $(-6,\ 3)$ minutes. Is the claim supported by this interval?
No, because the interval includes negative values, so the generic is definitely faster.
Yes, because the interval proves the two means are exactly equal.
No, because the interval includes positive values, so the brand is definitely faster.
No, because to support equality the interval must be centered at 0.
Yes, because 0 is in the interval, so equal means are plausible.
Explanation
This question tests understanding of claims about equality of means. The interval (-6, 3) for (μ_generic - μ_brand) contains 0, which means a difference of 0 (equal means) is plausible at the 95% confidence level. This supports the claim that the two medications have the same mean time to relief. Choices B and C incorrectly interpret the presence of negative or positive values as definitive evidence against equality. When testing a claim of equality between two means, we look for whether 0 is contained in the confidence interval. If 0 is included, the claim of equal means is supported; if 0 is excluded, the claim is not supported.
A company tests two website designs and measures mean time-on-page. They compute a 95% confidence interval for the difference in population means, $\mu_A-\mu_B$, as $(-12,\ 3)$ seconds. A manager claims: “Design A keeps users on the page longer on average than Design B.” Is the claim supported by the confidence interval?
No; because a 95% confidence interval means there is a 95% chance that $\mu_A=\mu_B$.
Yes; because the interval includes positive values, A must be better.
Yes; because the entire interval is below 0, so $\mu_A>\mu_B$.
Yes; because the interval proves A causes longer time-on-page.
No; because the interval includes 0, the data do not support $\mu_A>\mu_B$ at the 95% level.
Explanation
This question examines whether a confidence interval supports a directional claim about two means. The 95% confidence interval for μ_A - μ_B is (-12, 3) seconds, which includes both negative and positive values, crucially including zero. When a confidence interval for a difference contains zero, it means the data do not provide convincing evidence of a difference between the population means at the stated confidence level. Since zero is a plausible value for μ_A - μ_B, we cannot conclude that μ_A > μ_B. The manager's claim that Design A keeps users on the page longer is not supported because the interval suggests μ_A could be less than, equal to, or greater than μ_B. Remember: for a confidence interval to support μ₁ > μ₂, the entire interval for μ₁ - μ₂ must be positive.