AP Statistics Quiz: Justifying Claims Based On Confidence Interval

What this quiz covers

This quiz focuses on Justifying Claims Based On Confidence Interval, 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. Yes, because the entire interval is below 0, suggesting $p_X < p_Y$. (correct answer)
2. No, because the interval does not include 0, so there is no difference.
3. Yes, because the interval includes 0, so Brand X is lower.
4. No, because a negative interval means $p_X > p_Y$.
5. No, because the interval is for $p_Y - p_X$, not $p_X - p_Y$.

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.

Question 2

1. No, because the interval is negative, so $p_1$ is higher than $p_2$.
2. Yes, because the interval is entirely negative, indicating $p_1 - p_2 < 0$ and thus $p_2 > p_1$. (correct answer)
3. No, because $0$ is not in the interval, so there is no difference.
4. Yes, because the interval includes $0$, so Version 2 must be higher.
5. Yes, because the interval is for $p_2 - p_1$, and it is negative.

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.

Question 3

1. No, because the interval is too narrow to draw a conclusion.
2. Yes, because the interval is entirely positive, indicating $p_E > p_W$. (correct answer)
3. No, because 0 is not in the interval, so the proportions must be equal.
4. Yes, because the interval includes 0, so either region could be higher.
5. No, because the interval actually describes $p_W - p_E$.

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.

Question 4

1. Yes, because the interval contains positive values, so $p_1 > p_2$ is proven.
2. No, because the interval includes 0, so there is not convincing evidence that $p_1 > p_2$. (correct answer)
3. Yes, because the interval includes 0, so Design 1 must be better.
4. No, because the interval is entirely positive.
5. 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.

Question 5

1. Yes, because the entire interval is below 0, so $p_{North} < p_{South}$ is plausible. (correct answer)
2. No, because the interval is negative, which means $p_{North} > p_{South}$.
3. No, because the interval does not include 0, so there is no difference.
4. Yes, because the interval includes only negative values, so $p_{North} = p_{South}$.
5. No, because a confidence interval cannot indicate direction of a difference.

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.

Question 6

1. Yes, because the interval is positive, so $p_{Method2} > p_{Method1}$.
2. No, because the interval is entirely above 0, which supports $p_{Method1} > p_{Method2}$, not the claim. (correct answer)
3. Yes, because 0 is not in the interval, so Method 2 must be higher.
4. No, because 98% confidence is too high to compare proportions.
5. Yes, because the interval does not include 0, so either method could be higher.

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.

Question 7

1. Yes, because the interval includes 0, so School 1 must be higher.
2. No, because the interval includes 0, so there is not convincing evidence that $p_1 > p_2$. (correct answer)
3. Yes, because the interval is not entirely negative.
4. No, because the interval is entirely above 0.
5. 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.

Question 8

1. Yes, because the interval crosses 0, which confirms Clinic A is higher.
2. No, because the interval includes 0, so it does not provide convincing evidence that $p_A > p_B$. (correct answer)
3. Yes, because the interval includes positive values, so $p_A$ must be greater.
4. No, because the interval is entirely below 0.
5. Yes, because the interval is for $p_B - p_A$ and it includes negative values.

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.

Question 9

1. Yes, because the entire interval is below 0, suggesting $p_S < p_C$. (correct answer)
2. No, because the interval does not include 0, so there is no difference.
3. Yes, because the interval includes 0, so $p_S$ must be lower.
4. No, because a negative interval means the supplement group is higher.
5. No, because the interval is for $p_C - p_S$, not $p_S - p_C$.

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.

Question 10

1. Yes, because the interval contains positive values, so 2024 is higher.
2. No, because the interval includes 0, so higher approval in 2024 is not clearly supported. (correct answer)
3. Yes, because the interval includes 0, which confirms 2024 is higher.
4. No, because the negative lower bound proves 2024 must be higher.
5. 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").

Question 11

1. No, because the interval contains 0, so there may be no difference.
2. Yes, because the entire interval is above 0, so $p_9 > p_{10}$ is plausible. (correct answer)
3. No, because the interval is positive, which means $p_{10} > p_9$.
4. Yes, because 0 is not in the interval, so the proportions must be equal.
5. No, because a confidence interval cannot be used to compare two proportions.

Explanation: This question tests whether you can justify claims about comparing two proportions using a confidence interval. The confidence interval (0.03, 0.17) represents plausible values for p₉ - p₁₀, where p₉ is the proportion for Grade 9 and p₁₀ is the proportion for Grade 10. Since the entire interval is positive (above 0), this means p₉ - p₁₀ > 0, which is equivalent to p₉ > p₁₀. The claim that "Grade 9 has a higher proportion" is indeed supported because all plausible values for the difference show Grade 9 higher than Grade 10. When a confidence interval for a difference does not contain 0, we have evidence of a significant difference in the direction indicated by the interval's sign.

Question 12

1. Yes, because the interval includes negative values, so Brand A is higher.
2. Yes, because 0 is in the interval, so Brand A must be higher.
3. No, because the interval includes 0, so $p_A > p_B$ is not clearly supported. (correct answer)
4. No, because the interval is centered near 0, which proves $p_A = p_B$.
5. Yes, because the upper end is positive, so $p_A > p_B$ is guaranteed.

Explanation: This question asks whether a confidence interval supports a claim about Brand A having a higher proportion than Brand B. The 90% confidence interval (-0.08, 0.04) for p_A - p_B contains both negative and positive values, including 0. When 0 is in the confidence interval for a difference, it means that "no difference" (p_A = p_B) is a plausible value. Since the interval includes scenarios where p_A < p_B (negative values), p_A = p_B (zero), and p_A > p_B (positive values), we cannot conclusively support the claim that Brand A is higher. The presence of 0 in the interval indicates we lack sufficient evidence to claim one proportion is definitively higher than the other.

Question 13

1. No, because $0$ is not in the interval, so there is no difference between the grades.
2. Yes, because the entire interval is above $0$, so $p_9>p_{12}$ is plausible. (correct answer)
3. No, because the interval is above $0$, which indicates $p_{12}>p_9$.
4. Yes, because $0$ is not in the interval, so $p_9=p_{12}$ is most likely.
5. Not enough information; a confidence interval cannot be used to compare two proportions.

Explanation: This question tests understanding of confidence intervals for comparing two proportions. The interval (0.04, 0.18) represents plausible values for p₉ - p₁₂, where p₉ is the proportion of 9th graders and p₁₂ is the proportion of 12th graders who prefer online homework. Since the entire interval is positive (above 0), this means p₉ - p₁₂ > 0, which translates to p₉ > p₁₂. The student's claim that "a higher proportion of 9th graders than 12th graders prefer online homework" is indeed supported by the interval. When a confidence interval for a difference does not contain 0 and is entirely positive, it provides evidence that the first proportion is larger than the second.

Question 14

1. Yes, because the interval contains positive values, so $p_1>p_2$ is possible.
2. Yes, because the interval includes 0, which confirms no difference.
3. No, because the interval includes 0, so it does not provide convincing evidence that $p_1>p_2$. (correct answer)
4. No, because the interval is negative overall, so $p_2>p_1$.
5. Yes, but only if the interval had been calculated as $p_2-p_1$.

Explanation: This question requires understanding how the inclusion of 0 in a confidence interval affects hypothesis testing. The 90% confidence interval for p_1 - p_2 is (-0.005, 0.065), which includes 0. When a confidence interval for a difference includes 0, it means the true difference could be zero (no difference between the populations), positive ($p_1$ > $p_2$), or negative ($p_1$ < $p_2$). Therefore, the interval does not provide convincing evidence to support the manager's claim that Design 1 has a higher purchase rate than Design 2. The presence of 0 in the interval indicates that we cannot rule out the possibility of no difference or even Design 2 being better. To support a directional claim like p_1 > p_2, the entire interval must be above 0.

Question 15

1. Yes, because $p_H-p_S$ is negative throughout the interval, so $p_S>p_H$ is supported. (correct answer)
2. No, because the interval is negative, so $p_H>p_S$ is supported.
3. Yes, because $0$ is not in the interval, so there is no difference.
4. No, because $0$ is not in the interval, so the claim cannot be evaluated.
5. Not enough information; the confidence level would need to be 90% to compare.

Explanation: This question requires careful interpretation of a negative interval and the order of subtraction. The 95% confidence interval for p_H - p_S is (-0.22, -0.05), which is entirely negative. Since the interval is entirely below 0, this means p_H - p_S < 0, which translates to p_H < p_S, or equivalently, p_S > p_H. The advisor claims "A higher proportion of students accept STEM offers than Humanities offers," which means p_S > p_H. This claim is indeed supported by the interval. When a confidence interval for a difference is entirely negative, it indicates the second proportion (the one being subtracted) is larger than the first, which in this case supports the claim about STEM having higher acceptance rates.

Question 16

1. No, because the interval includes 0, so there may be no difference.
2. Yes, because the entire interval is above 0, so $p_A>p_B$ is plausible. (correct answer)
3. No, because the interval is positive, which means $p_B>p_A$.
4. Yes, because the interval includes 0, which proves the programs are equal.
5. No, because the interval is for $p_B-p_A$, not for $p_A-p_B$.

Explanation: This question tests your ability to interpret a confidence interval for the difference in proportions to justify a claim. The 95% confidence interval for p_A - p_B is (0.02, 0.22), which means we are 95% confident that the true difference between Program A's pass rate and Program B's pass rate lies between 0.02 and 0.22. Since the entire interval is above 0 (all positive values), this provides convincing evidence that p_A > p_B, meaning Program A has a higher pass rate than Program B. The key insight is that when a confidence interval for p_1 - p_2 contains only positive values, it supports the claim that p_1 > p_2. Choice C incorrectly interprets positive values as supporting p_B > p_A, which would be true if the interval were for p_B - p_A instead.

Question 17

1. Yes, because the interval is negative, so $p_C>p_D$.
2. No, because the interval is negative, suggesting $p_C<p_D$. (correct answer)
3. Yes, because $0$ is not in the interval, so Team C and Team D are equal.
4. No, because $0$ is not in the interval, so there is no evidence of a difference.
5. Yes, because the interval does not include $0$, which always means the first team is better.

Explanation: This question examines interpretation of a negative confidence interval. The 95% confidence interval for p_C - p_D is (-0.12, -0.01), which is entirely negative (below 0). When a confidence interval for a difference is entirely negative, it means p_C - p_D < 0, which translates to p_C < p_D. The fan claims "Team C has a higher home-win proportion than Team D," which would require p_C > p_D. However, the interval shows the opposite: Team D has a higher home-win proportion than Team C. The claim is not supported; in fact, it is contradicted by the interval. Remember that when interpreting confidence intervals for differences, negative values indicate the first quantity is smaller than the second, not larger.

Question 18

1. Yes, because the interval contains 0, which proves no difference and therefore higher retention for Program 1.
2. No, because the interval includes 0, so it does not provide convincing evidence that $p_1>p_2$. (correct answer)
3. Yes, because the upper endpoint is positive, so $p_1>p_2$ is guaranteed.
4. No, because the interval is for $p_2-p_1$, which reverses the conclusion.
5. Yes, because the interval includes negative values, so $p_1>p_2$.

Explanation: This question requires careful analysis of a confidence interval that includes zero. The 90% confidence interval for p_1 - p_2 is (-0.01, 0.07), where p_1 is the retention rate for Program 1 and p_2 for Program 2. The administrator claims Program 1 has a higher retention rate ($p_1$ > $p_2$), but the interval includes negative values (as low as -0.01), zero, and positive values (up to 0.07). This means the true difference could be negative (Program 2 better), zero (no difference), or positive (Program 1 better). Since we cannot rule out the possibilities that p_1 ≤ p_2, the interval does not provide convincing evidence to support the claim. For a confidence interval to support a directional claim, it must exclude 0 and be entirely on one side of 0.

Question 19

1. No, because 0 is not in the interval, so there is no difference between the grades.
2. Yes, because the entire interval is above 0, suggesting $p_9 > p_{10}$. (correct answer)
3. No, because the interval includes only positive values, which means Grade 10 must be higher.
4. Yes, because the interval contains 0, so either grade could be higher.
5. No, because the interval is for $p_{10} - p_9$, not $p_9 - p_{10}$.

Explanation: This question assesses the skill of justifying claims using confidence intervals for the difference in proportions, specifically whether the interval supports a claim that one proportion is higher than another. The 95% confidence interval for p₉ - p₁₀ is (0.04, 0.18), which is entirely positive and does not include zero, indicating that the true difference is likely positive and thus p₉ > p₁₀. This supports the student's claim that Grade 9 has a higher proportion preferring online homework than Grade 10. A common distractor is choice A, which incorrectly states that no difference exists because zero is not in the interval, misunderstanding that exclusion of zero supports a difference. In a mini-lesson on comparative claims: when evaluating if p₁ > p₂ using a CI for p₁ - p₂, check if the entire interval is above zero; if yes, the claim is supported at that confidence level. If the interval includes zero, there's insufficient evidence for the claim. Remember to note the order of subtraction, as reversing it would flip the interpretation.

Question 20

1. No, because the interval is above 0, which means Democrats are higher.
2. Yes, because the entire interval is greater than 0, indicating $p_I > p_D$. (correct answer)
3. No, because 0 is not in the interval, so we cannot compare the groups.
4. Yes, because the interval includes 0, so the claim must be true.
5. No, because the interval is for $p_D - p_I$, not $p_I - p_D$.

Explanation: The skill is using confidence intervals to justify claims about voter proportion differences, evaluating if independents favor the candidate more. The 95% CI for p_I - p_D is (0.01, 0.25), fully positive excluding zero, supporting p_I > p_D and the commentator's claim. This indicates independents likely have higher favor. Choice A distracts by misinterpreting positive interval as Democrats higher, reversing the subtraction. Mini-lesson: a positive-only CI for p₁ - p₂ supports p₁ > p₂; if zero is in, evidence is weak for inequality. Always check which group is subtracted to align with the claim. Polling intervals account for sampling variability in proportions.