Justifying Claims Based on Confidence Interval
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AP Statistics › Justifying Claims Based on Confidence Interval
Two brands of batteries are tested for the proportion that last at least 10 hours. A 95% confidence interval for the difference in proportions, defined as $p_{X} - p_{Y}$ (Brand X minus Brand Y), is $(-0.20, -0.05)$. A consumer claims, “Brand X has a lower proportion lasting at least 10 hours than Brand Y.” Is the claim supported by the confidence interval?
Yes, because the entire interval is below 0, suggesting $p_X < p_Y$.
No, because the interval does not include 0, so there is no difference.
No, because a negative interval means $p_X > p_Y$.
No, because the interval is for $p_Y - p_X$, not $p_X - p_Y$.
Yes, because the interval includes 0, so Brand X is lower.
Explanation
The skill here is justifying claims based on confidence intervals for proportion differences, determining if the interval supports a claim that one brand has a lower durability proportion than another. The 95% CI for p_X - p_Y is (-0.20, -0.05), entirely negative and excluding zero, suggesting p_X < p_Y, which supports the consumer's claim. This means Brand X likely has a lower proportion lasting at least 10 hours. Distractor choice D wrongly claims a negative interval means p_X > p_Y, confusing the sign of the difference. Mini-lesson: to claim p₁ < p₂ using CI for p₁ - p₂, the interval must be entirely below zero; if it straddles zero, no strong evidence for inequality. Pay attention to which proportion is subtracted to avoid reversing conclusions. This approach aligns with hypothesis testing at the corresponding significance level.
A company tests two versions of an email subject line and records the proportion of recipients who open the email. Version 1 was sent to 400 recipients and 112 opened it; Version 2 was sent to 380 recipients and 133 opened it. A 99% confidence interval for $p_1 - p_2$ is $(-0.15,\ -0.02)$. A marketing analyst claims that Version 2 has a higher open rate than Version 1. Is the claim supported by the confidence interval?
No, because $0$ is not in the interval, so there is no difference.
Yes, because the interval is for $p_2 - p_1$, and it is negative.
Yes, because the interval is entirely negative, indicating $p_1 - p_2 < 0$ and thus $p_2 > p_1$.
Yes, because the interval includes $0$, so Version 2 must be higher.
No, because the interval is negative, so $p_1$ is higher than $p_2$.
Explanation
The skill involves using a confidence interval for p1 - p2 to evaluate claims about email open rates. The 99% interval (-0.15, -0.02) is entirely below 0, supporting p1 < p2, which aligns with the claim that Version 2 has a higher rate (p2 > p1). This means all plausible differences are negative, confirming the claim. Choice A is a distractor, misstating that a negative interval means p1 > p2, confusing the sign's implication. Mini-lesson on comparative claims: An entirely negative CI for p1 - p2 justifies p1 < p2 (or p2 > p1), while positive supports p1 > p2, and including 0 suggests insufficient evidence for inequality. Sample rates (112/400 = 0.28 for 1, 133/380 ≈ 0.35 for 2) match this finding.
A streaming service compares the proportion of users who watch documentaries among subscribers in Region East versus Region West. A 95% confidence interval for $p_E - p_W$ is $(0.08, 0.11)$. An analyst claims, “A higher proportion of subscribers in the East watch documentaries than in the West.” Is the claim supported by the confidence interval?
No, because 0 is not in the interval, so the proportions must be equal.
Yes, because the interval includes 0, so either region could be higher.
Yes, because the interval is entirely positive, indicating $p_E > p_W$.
No, because the interval actually describes $p_W - p_E$.
No, because the interval is too narrow to draw a conclusion.
Explanation
This question focuses on the skill of justifying claims with confidence intervals for proportion differences, assessing if the interval supports higher documentary watching in one region. The 95% CI for p_E - p_W is (0.08, 0.11), entirely positive without zero, supporting that p_E > p_W and thus the analyst's claim. This suggests East subscribers likely have a higher proportion. Distractor choice D misstates that including zero allows either to be higher, but here zero is excluded. Mini-lesson: a claim of p₁ > p₂ is backed if the CI for p₁ - p₂ is wholly positive; zero inclusion implies insufficient evidence for superiority. Ensure the difference definition aligns with the claim to interpret correctly. Higher confidence levels widen intervals, making claims harder to support.
A company tests two website designs for the proportion of visitors who make a purchase: Design 1 and Design 2. A 90% confidence interval for the difference $p_1 - p_2$ is $(-0.02, 0.07)$. A designer claims, “Design 1 leads to a higher purchase rate than Design 2.” Is the claim supported by the confidence interval?
Yes, because the interval contains positive values, so $p_1 > p_2$ is proven.
Yes, because the interval includes 0, so Design 1 must be better.
No, because the interval includes 0, so there is not convincing evidence that $p_1 > p_2$.
No, because the interval is entirely positive.
Yes, because the subtraction should be $p_2 - p_1$, which would make the interval positive.
Explanation
This question tests justifying claims via confidence intervals for purchase proportion differences, checking if one design yields higher rates. The 90% CI for p₁ - p₂ is (-0.02, 0.07), including zero, so not convincing evidence that p₁ > p₂, unsupported claim. The interval allows for Design 1 being slightly better, worse, or equal. Distractor choice D incorrectly notes the interval as entirely positive, but it's not. Mini-lesson: to justify p₁ > p₂, ensure the CI for p₁ - p₂ excludes zero and is positive throughout; zero inclusion suggests no clear winner. Confirm the difference calculation fits the claim. Lower confidence levels narrow intervals, potentially supporting claims more easily.
A city surveys residents in the North and South districts about supporting a new park. A 95% confidence interval for the difference in proportions, defined as $p_{North} - p_{South}$, is $(-0.21,\ -0.05)$. A council member claims, “The North district has a lower proportion supporting the park than the South district.” Is the claim supported by the confidence interval?
No, because the interval does not include 0, so there is no difference.
No, because the interval is negative, which means $p_{North} > p_{South}$.
Yes, because the interval includes only negative values, so $p_{North} = p_{South}$.
No, because a confidence interval cannot indicate direction of a difference.
Yes, because the entire interval is below 0, so $p_{North} < p_{South}$ is plausible.
Explanation
This question tests understanding of negative confidence intervals for differences in proportions. The 95% confidence interval (-0.21, -0.05) for p_North - p_South is entirely negative, meaning all plausible values for the difference are below 0. When p_North - p_South < 0, this is equivalent to p_North < p_South, which means the North district has a lower proportion than the South district. The claim that "The North district has a lower proportion supporting the park than the South district" is supported by the interval. Students often confuse the direction when intervals are negative, but remember: if the difference A - B is negative, then A is smaller than B.
Two different study methods are compared for the proportion of students who earn an A on a final exam. A 98% confidence interval for the difference in proportions, defined as $p_{Method1} - p_{Method2}$, is $(0.05,\ 0.20)$. A teacher claims, “Method 2 results in a higher A rate than Method 1.” Is the claim supported by the confidence interval?
Yes, because the interval does not include 0, so either method could be higher.
Yes, because 0 is not in the interval, so Method 2 must be higher.
No, because 98% confidence is too high to compare proportions.
Yes, because the interval is positive, so $p_{Method2} > p_{Method1}$.
No, because the interval is entirely above 0, which supports $p_{Method1} > p_{Method2}$, not the claim.
Explanation
This question requires careful attention to the order of subtraction in the confidence interval. The 98% confidence interval (0.05, 0.20) is for p_Method1 - p_Method2, and it's entirely positive. This means p_Method1 - p_Method2 > 0, which translates to p_Method1 > p_Method2. However, the claim states that "Method 2 results in a higher A rate than Method 1," which would require p_Method2 > p_Method1. The confidence interval actually supports the opposite of the claim—it shows Method 1 has a higher A rate than Method 2. This is a common trap where students must carefully match the order of subtraction in the interval with the direction of the claim being made.
A researcher compares the proportion of left-handed students in School 1 and School 2. A 99% confidence interval for the difference in proportions, defined as $p_1 - p_2$, is $(0.00, 0.09)$. The researcher claims, “School 1 has a higher proportion of left-handed students than School 2.” Is the claim supported by the confidence interval?
No, because the interval is entirely above 0.
No, because the interval includes 0, so there is not convincing evidence that $p_1 > p_2$.
Yes, because the interval is not entirely negative.
Yes, because the interval includes 0, so School 1 must be higher.
Yes, because the interval proves $p_2 - p_1$ is positive.
Explanation
This problem evaluates the skill of using confidence intervals to justify claims about proportion differences, specifically if the interval supports one school having more left-handed students. The 99% CI for p₁ - p₂ is (0.00, 0.09), which includes zero, indicating no convincing evidence that p₁ > p₂ since zero (no difference) is plausible. Therefore, the researcher's claim is not supported. Choice A is a distractor, incorrectly assuming including zero proves p₁ > p₂, which it doesn't. Mini-lesson on comparative claims: for p₁ > p₂, the CI for p₁ - p₂ must be fully above zero; including zero means the data don't rule out equality. Note the confidence level affects interval width but not the zero-check principle. Always confirm the subtraction order matches the claim.
A hospital compares the proportion of patients who report being “very satisfied” after treatment at Clinic A versus Clinic B. A 95% confidence interval for the difference in proportions, $p_A - p_B$, is $(-0.06, 0.14)$. A manager claims, “Clinic A has a higher ‘very satisfied’ rate than Clinic B.” Is the claim supported by the confidence interval?
Yes, because the interval includes positive values, so $p_A$ must be greater.
Yes, because the interval is for $p_B - p_A$ and it includes negative values.
No, because the interval includes 0, so it does not provide convincing evidence that $p_A > p_B$.
Yes, because the interval crosses 0, which confirms Clinic A is higher.
No, because the interval is entirely below 0.
Explanation
The skill assessed is justifying claims using confidence intervals for differences in satisfaction proportions, determining if one clinic rates higher. The 95% CI for p_A - p_B is (-0.06, 0.14), including zero, so no convincing evidence for p_A > p_B, not supporting the manager's claim. Plausible differences include negative values, zero, or positive. Choice C distracts by saying positive values prove p_A > p_B, ignoring zero and negatives. Mini-lesson: support for p₁ > p₂ requires the CI for p₁ - p₂ to be entirely above zero; straddling zero means the claim lacks strong backing. Verify the subtraction order to match the comparison. This method equates to not rejecting the null of no difference when zero is included.
A nutrition study compares the proportion of adults who meet a daily fiber goal in Group S (supplement) versus Group C (control). A 98% confidence interval for $p_S - p_C$ is $(-0.15, -0.01)$. A researcher claims, “The supplement group has a lower proportion meeting the fiber goal than the control group.” Is the claim supported by the confidence interval?
Yes, because the interval includes 0, so $p_S$ must be lower.
No, because the interval is for $p_C - p_S$, not $p_S - p_C$.
Yes, because the entire interval is below 0, suggesting $p_S < p_C$.
No, because a negative interval means the supplement group is higher.
No, because the interval does not include 0, so there is no difference.
Explanation
This problem assesses justifying claims with confidence intervals for fiber goal proportions, determining if the supplement group performs worse. The 98% CI for p_S - p_C is (-0.15, -0.01), entirely negative without zero, supporting p_S < p_C and the researcher's claim. The supplement group likely has a lower proportion meeting the goal. Distractor choice D wrongly says negative means supplement higher, ignoring the sign. Mini-lesson: for claiming p₁ < p₂, the CI for p₁ - p₂ should be fully negative; including zero doesn't support the inequality. Match the difference order to the claim for accurate interpretation. Higher confidence like 98% widens the interval, requiring stronger evidence.
A survey compares the proportion of adults who approve of a policy in 2024 versus 2025. A 95% confidence interval for the difference in proportions, defined as $p_{2024} - p_{2025}$, is $( -0.04,\ 0.10)$. A reporter claims, “Approval was higher in 2024 than in 2025.” Is the claim supported by the confidence interval?
No, because the negative lower bound proves 2024 must be higher.
Yes, because the interval includes 0, which confirms 2024 is higher.
Yes, because the interval contains positive values, so 2024 is higher.
No, because the interval includes 0, so higher approval in 2024 is not clearly supported.
Yes, because a 95% interval always supports the claim in the subtraction order.
Explanation
This question tests understanding of confidence intervals that span both negative and positive values. The 95% confidence interval (-0.04, 0.10) for p_2024 - p_2025 includes negative values, zero, and positive values. The presence of 0 in the interval means "no difference in approval" is a plausible scenario. Additionally, the negative values suggest that approval could have been lower in 2024 than in 2025. Since the interval includes cases where p_2024 ≤ p_2025, the claim that "Approval was higher in 2024 than in 2025" is not clearly supported. For a directional claim to be supported, the entire confidence interval must exclude 0 and be on the appropriate side (positive for "higher," negative for "lower").