AP Statistics Quiz: Confidence Intervals Difference Of Two Proportions

What this quiz covers

This quiz focuses on Confidence Intervals Difference Of Two Proportions, 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. There is a 95% chance that the difference $p_1 - p_2$ is between 0.01 and 0.11.
2. We are 95% confident that Version 2's 2FA proportion is 0.01 to 0.11 higher than Version 1's 2FA proportion.
3. We are 95% confident that the proportion of users who enable 2FA is between 0.01 and 0.11 for Version 1.
4. We are 95% confident that Version 1's 2FA proportion is 0.01 to 0.11 higher than Version 2's 2FA proportion. (correct answer)
5. Because 0 is not in the interval, the sample proportions must be equal.

Explanation: This question tests interpretation of a positive confidence interval. The interval (0.01, 0.11) for p₁ - p₂ indicates Version 1 has a higher 2FA enablement proportion than Version 2. Choice D correctly states that we are 95% confident Version 1's 2FA proportion is 0.01 to 0.11 higher than Version 2's. Choice A incorrectly treats confidence as probability. Choice B reverses which version is higher. Choice C only describes one proportion. Choice E incorrectly concludes the samples are equal when the interval doesn't contain 0.

Question 2

1. We are 95% confident that $p_2 - p_1$ is between 0.04 and 0.20.
2. If we repeated the sampling many times, 95% of the intervals would contain the sample difference $\hat p_1 - \hat p_2$.
3. We are 95% confident that the proportion of adults who drink coffee daily is between 0.04 and 0.20 in Region 1.
4. We are 95% confident that Region 1's daily-coffee proportion is 0.04 to 0.20 higher than Region 2's daily-coffee proportion. (correct answer)
5. There is a 95% probability that the true difference $p_1 - p_2$ is outside the interval $(0.04, 0.20)$.

Explanation: This question involves interpreting a confidence interval for p₁ - p₂, where p₁ is the proportion of all adults in Region 1 who drink coffee daily and p₂ is the proportion in Region 2. The interval (0.04, 0.20) is entirely positive, indicating Region 1 has a higher proportion. Choice D correctly states that we are 95% confident Region 1's proportion is 0.04 to 0.20 higher than Region 2's proportion. Choice A reverses the order of subtraction. Choice B misunderstands what the interval estimates. Choice C only describes one proportion, not the difference. Choice E incorrectly states the probability is outside the interval.

Question 3

1. We are 95% confident that Candidate A's approval proportion in County 1 is 0.01 to 0.11 higher than in County 2. (correct answer)
2. There is a 95% probability that the interval $(0.01, 0.11)$ will contain $\hat p_1 - \hat p_2$.
3. Because 0 is not in the interval, we know for sure that $p_1 - p_2 = 0.06$.
4. We are 95% confident that between 1% and 11% of all voters approve of Candidate A in County 1.
5. We are 95% confident that Candidate A's approval proportion in County 2 is 0.01 to 0.11 higher than in County 1.

Explanation: This question involves interpreting a confidence interval for p₁ - p₂, where p₁ is Candidate A's approval proportion in County 1 and p₂ is the approval proportion in County 2. The interval (0.01, 0.11) is entirely positive, indicating County 1 has higher approval. Choice A correctly states that we are 95% confident Candidate A's approval proportion in County 1 is 0.01 to 0.11 higher than in County 2. Choice B misunderstands what the interval estimates. Choice C incorrectly assumes we know the exact difference. Choice D confuses the difference with a single proportion. Choice E reverses the counties.

Question 4

1. We are 98% confident that the true difference in passing rates, $p_1 - p_2$, is between 0.02 and 0.30. (correct answer)
2. There is a 98% chance that Session 1 will cause a student to pass the exam.
3. We are 98% confident that $p_2 - p_1$ is between 0.02 and 0.30.
4. Because the interval does not include 0, there is no difference between the two sessions in the population.
5. We are 98% confident that between 2% and 30% of all students pass the exam.

Explanation: This question tests understanding of confidence intervals for differences in proportions. The interval (0.02, 0.30) for p₁ - p₂ estimates the difference in passing rates between Session 1 and Session 2. Since the interval is entirely positive, Session 1 has a higher passing rate. Choice A correctly interprets this: we are 98% confident that the true difference p₁ - p₂ is between 0.02 and 0.30. Choice B incorrectly implies causation. Choice C reverses the order of subtraction. Choice D misinterprets a non-zero interval. Choice E confuses the difference with individual proportions.

Question 5

1. We are 90% confident that the proportion of renters who support the fee is between 0.04 and 0.18 lower than the proportion of homeowners who support the fee. (correct answer)
2. There is a 90% probability that $p_R - p_H$ equals a value between $-0.18$ and $-0.04$.
3. Because the interval is negative, 90% of renters and 90% of homeowners support the fee.
4. We are 90% confident that $p_H - p_R$ is between $-0.18$ and $-0.04$.
5. Since 0 is not in the interval, there is no difference between renters and homeowners in the population.

Explanation: This question involves interpreting a confidence interval for p_R - p_H, where p_R is the proportion of all renters who support the fee and p_H is the proportion of all homeowners who support the fee. The interval (-0.18, -0.04) is entirely negative, meaning p_R is less than p_H. Choice A correctly states that we are 90% confident the proportion of renters who support the fee is between 0.04 and 0.18 lower than the proportion of homeowners. Choice B incorrectly treats confidence as probability. Choice C completely misinterprets the negative interval. Choice D has the wrong order of subtraction (it would give a positive interval). Choice E incorrectly concludes no difference when the interval doesn't contain 0.

Question 6

1. There is a 98% chance that the true proportions $p_N$ and $p_S$ will change so that their difference stays between 0.01 and 0.16.
2. We are 98% confident that the North's support proportion is between 0.01 and 0.16 higher than the South's support proportion. (correct answer)
3. We are 98% confident that the South's support proportion is between 0.01 and 0.16 higher than the North's support proportion.
4. Because 0 is not in the interval, the probability that a randomly selected voter supports the measure is between 0.01 and 0.16.
5. 98% of the time, the sample difference $\hat p_N - \hat p_S$ will be between 0.01 and 0.16 for these same samples.

Explanation: This question tests the skill of interpreting a confidence interval for pN - pS, the difference in voter support proportions between regions. The 98% interval (0.01, 0.16) indicates we are 98% confident that the North's proportion is between 0.01 and 0.16 higher than the South's. Choice C distracts by reversing which region is higher, contradicting the positive interval. Choice A incorrectly suggests the proportions themselves change within the interval. For a mini-lesson: confidence intervals for differences rely on normal approximations for large samples, giving a range where pN - pS plausibly falls. Excluding 0 with positive endpoints provides evidence of higher support in the North.

Question 7

1. Because 0 is in the interval, we are 95% confident that County X has a lower flu-shot proportion than County Y.
2. We are 95% confident that the true difference in flu-shot proportions $(p_X - p_Y)$ is between $-0.06$ and $0.12$. (correct answer)
3. There is a 95% probability that the true difference $(p_X - p_Y)$ is exactly 0.
4. We are 95% confident that 6% to 12% of adults in County X received a flu shot.
5. If we took many samples, 95% of adults would fall within $-0.06$ and $0.12$ of being vaccinated.

Explanation: This question involves interpreting a confidence interval containing zero for flu shot proportions. The interval (-0.06, 0.12) for (pₓ - pᵧ) includes both negative and positive values, indicating uncertainty about which county has higher vaccination rates. Choice B correctly states we are 95% confident that the true difference in flu-shot proportions is between -0.06 and 0.12. Choice A incorrectly concludes County X has lower rates when positive values in the interval suggest it could be higher. Choice C incorrectly assigns probability to exact equality. Choice D misinterprets the interval as being about a single proportion. Choice E makes no statistical sense. When zero is in the interval, we cannot determine which population proportion is larger.

Question 8

1. There is a 95% probability that the true difference $(p_9 - p_{10})$ is between 0.02 and 0.20.
2. We are 95% confident that the true difference in proportions $(p_9 - p_{10})$ is between 0.02 and 0.20. (correct answer)
3. About 95% of ninth-graders prefer online homework, and about 95% of tenth-graders do too.
4. Because 0 is not in the interval, there is no difference between $p_9$ and $p_{10}$.
5. We are 95% confident that the difference in sample proportions $(\hat p_9 - \hat p_{10})$ is between 0.02 and 0.20.

Explanation: This question tests understanding of confidence interval interpretation for the difference of two proportions. The interval (0.02, 0.20) estimates the true difference in population proportions (p₉ - p₁₀). Choice B correctly states we are 95% confident that the true difference in proportions is between 0.02 and 0.20. Choice A incorrectly uses probability language - confidence intervals don't give probabilities about parameters. Choice C misinterprets the interval as being about individual proportions rather than their difference. Choice D incorrectly concludes no difference when 0 is NOT in the interval. Choice E incorrectly refers to sample proportions rather than population proportions. Remember: confidence intervals estimate population parameters, not sample statistics.

Question 9

1. We are 90% confident that Design A's purchase proportion is between $-0.18$ and $-0.02$.
2. Because the interval is negative, we are 90% confident that $p_B$ is between 0.02 and 0.18 greater than $p_A$.
3. There is a 90% chance that $(p_A - p_B)$ is negative for this experiment.
4. We are 90% confident that the true difference in purchase proportions $(p_A - p_B)$ is between $-0.18$ and $-0.02$. (correct answer)
5. Since 0 is not in the interval, the two sample proportions must be equal.

Explanation: This question involves interpreting a negative confidence interval for the difference of two proportions. The interval (-0.18, -0.02) estimates (pₐ - pᵦ), where negative values indicate Design A has a lower purchase proportion than Design B. Choice D correctly interprets this as being 90% confident that the true difference in purchase proportions is between -0.18 and -0.02. Choice A incorrectly refers to a single proportion rather than the difference. Choice B correctly notes that pᵦ is greater than pₐ but reverses the order of subtraction. Choice C incorrectly uses probability language about the parameter. Choice E incorrectly concludes equality when 0 is NOT in the interval. When interpreting negative intervals, pay attention to which proportion is subtracted from which.

Question 10

1. We are 99% confident that the true difference in support proportions $(p_1 - p_2)$ is between $-0.05$ and $0.13$. (correct answer)
2. There is a 99% probability that $p_1 = p_2$ because 0 is in the interval.
3. Because 0 is in the interval, we are 99% confident that Neighborhood 1 has a higher support proportion than Neighborhood 2.
4. We are 99% confident that the sample difference $(\hat p_1 - \hat p_2)$ is between $-0.05$ and $0.13$ for all samples.
5. About 99% of all residents in both neighborhoods support the park.

Explanation: This question tests interpretation of a confidence interval that contains zero. The interval (-0.05, 0.13) for (p₁ - p₂) includes both negative and positive values, indicating uncertainty about which neighborhood has higher support. Choice A correctly states we are 99% confident that the true difference in support proportions is between -0.05 and 0.13. Choice B incorrectly assigns probability to the equality of parameters. Choice C incorrectly concludes Neighborhood 1 has higher support when the interval includes negative values. Choice D incorrectly refers to sample differences rather than population differences. Choice E completely misinterprets the interval as being about individual proportions. When zero is in the interval, we cannot conclude which population proportion is larger.

Question 11

1. We are 92% confident that the true difference in 4-year graduation proportions $(p_A - p_B)$ is between 0.01 and 0.17. (correct answer)
2. There is a 92% chance that Program A has a higher graduation proportion than Program B.
3. We are 92% confident that between 1% and 17% of students in Program A graduate in 4 years.
4. Because 0 is not in the interval, the true difference $(p_A - p_B)$ must be 0.09 (the midpoint).
5. We are 92% confident that the difference in sample proportions equals a value between 0.01 and 0.17 for all random samples of these sizes.

Explanation: This question tests proper interpretation of a positive confidence interval for graduation rates. The interval (0.01, 0.17) estimates (pₐ - pᵦ), indicating Program A has a higher 4-year graduation proportion. Choice A correctly states we are 92% confident that the true difference in 4-year graduation proportions is between 0.01 and 0.17. Choice B incorrectly uses probability language about which program is better. Choice C misinterprets the interval as being about a single proportion. Choice D incorrectly assumes the true difference must be the midpoint. Choice E incorrectly refers to sample proportions across all possible samples. Since 0 is not in the interval, we can conclude Program A has a higher graduation rate than Program B.

Question 12

1. There is a 95% probability that Player 1's true free-throw proportion is between $-0.01$ and $0.17$.
2. We are 95% confident that the true difference in free-throw proportions $(p_1 - p_2)$ is between $-0.01$ and $0.17$. (correct answer)
3. Because 0 is in the interval, Player 2 definitely has a higher true free-throw proportion than Player 1.
4. We are 95% confident that Player 1 makes between 1% fewer and 17% more free throws than Player 2 in the sample.
5. If the sampling were repeated many times, 95% of the computed intervals would not contain $(p_1 - p_2)$.

Explanation: This question addresses interpretation of a confidence interval containing zero for basketball free throws. The interval (-0.01, 0.17) for (p₁ - p₂) includes both negative and positive values, though mostly positive. Choice B correctly states we are 95% confident that the true difference in free-throw proportions is between -0.01 and 0.17. Choice A incorrectly suggests a negative proportion is possible. Choice C incorrectly concludes Player 2 is better when the interval is mostly positive. Choice D incorrectly refers to sample proportions. Choice E misunderstands the meaning of confidence level. Since zero is in the interval, we cannot definitively conclude which player has a higher true free-throw proportion.

Question 13

1. There is a 95% chance that the true difference $p_9 - p_{10}$ is between 0.02 and 0.20.
2. We are 95% confident that the proportion of all ninth-graders who prefer online homework is between 0.02 and 0.20 higher than the proportion of all tenth-graders who prefer online homework. (correct answer)
3. About 95% of ninth-graders prefer online homework, and the difference from tenth-graders is between 0.02 and 0.20.
4. Since 0 is not in the interval, exactly 95% of ninth-graders prefer online homework more than tenth-graders.
5. We are 95% confident that the proportion of all tenth-graders who prefer online homework is between 0.02 and 0.20 higher than the proportion of all ninth-graders who prefer online homework.

Explanation: This question assesses the skill of interpreting a confidence interval for the difference of two proportions, specifically for the difference p9 - p10 in preferences for online homework. The 95% confidence interval (0.02, 0.20) indicates that we are 95% confident the true proportion of ninth-graders preferring online homework is between 0.02 and 0.20 higher than that of tenth-graders. A common distractor, like choice A, mistakenly treats the interval as a probability for the parameter rather than a confidence statement about the method capturing the true difference. Another distractor, choice E, reverses the order of subtraction, implying tenth-graders have a higher proportion, which contradicts the positive interval. In a mini-lesson on confidence intervals for differences: these intervals are calculated as (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}, providing a range for the plausible values of p1 - p2. Since the interval is entirely positive and excludes 0, there is evidence that ninth-graders have a higher preference rate in the population.

Question 14

1. We are 95% confident that Light A's germination proportion is between 0.22 and 0.01 lower than Light B's germination proportion. (correct answer)
2. There is a 95% probability that the sample difference $\hat p_A - \hat p_B$ is between $-0.22$ and $-0.01$.
3. We are 95% confident that Light A's germination proportion is between 0.22 and 0.01 higher than Light B's germination proportion.
4. Because the entire interval is negative, Light A must have a higher germination proportion than Light B.
5. 95% of individual seeds have a probability of germinating between $-0.22$ and $-0.01$ under Light A compared with Light B.

Explanation: This question focuses on interpreting a confidence interval for pA - pB, the difference in seed germination proportions under two lights. The 95% interval (-0.22, -0.01) means we are 95% confident that Light A's proportion is between 0.22 and 0.01 lower than Light B's. Choice C distracts by claiming 'higher' instead of 'lower,' ignoring the negative signs. Choice B wrongly applies the interval to sample differences rather than population parameters. Mini-lesson: confidence intervals for differences use the formula (\hat{p}_A - \hat{p}_B) \pm z^* \times SE, estimating where pA - pB lies with a given confidence. Since the interval is entirely negative, excluding 0, it provides evidence that Light B has a higher germination rate in the population.

Question 15

1. Because the interval includes 0, there is no difference between the two designs in the populations.
2. We are 99% confident that Design 1's purchase rate is between 0.05 lower and 0.01 higher than Design 2's purchase rate. (correct answer)
3. There is a 99% chance that $p_1 - p_2$ equals 0 since 0 is in the interval.
4. We are 99% confident that $p_2 - p_1$ is between $-0.05$ and 0.01.
5. 99% of all visitors will have purchase rates between $-0.05$ and 0.01 when comparing the two designs.

Explanation: This question tests interpreting a confidence interval for the difference in purchase rates, p1 - p2, between two website designs. The 99% interval (-0.05, 0.01) suggests we are 99% confident that Design 1's rate is between 0.05 lower and 0.01 higher than Design 2's. A frequent distractor, like choice A, claims that including 0 means no population difference, but it actually means no strong evidence against equality. Choice D incorrectly states the interval for p2 - p1 without adjusting the bounds properly. Mini-lesson: to form a CI for p1 - p2, use the sample difference plus/minus a critical value times the standard error; the interval captures plausible differences, and including 0 indicates the data are consistent with no difference. Here, the endpoints straddle 0, so we cannot conclude one design is superior.

Question 16

1. We are 95% confident that the proportion of engineering students who report high stress is between 0.03 and 0.21.
2. We are 95% confident that engineering's high-stress proportion is between 0.03 and 0.21 higher than business's high-stress proportion. (correct answer)
3. There is a 95% chance that $p_{eng} - p_{bus}$ is not between 0.03 and 0.21.
4. Because 0 is not in the interval, business students must have a higher high-stress proportion than engineering students.
5. 95% of all possible samples will produce a confidence interval of $(0.03,\ 0.21)$.

Explanation: This question focuses on interpreting a confidence interval for p_eng - p_bus, the difference in high-stress proportions between majors. The 95% interval (0.03, 0.21) means we are 95% confident that engineering's proportion is between 0.03 and 0.21 higher than business's. Choice D distracts by reversing which major has higher stress, ignoring the positive interval. Choice C misstates the probability as the chance the difference is not in the interval. In a mini-lesson: calculate CIs for differences using sample proportions and critical values; they estimate population differences reliably. Positive endpoints excluding 0 indicate engineering students likely experience higher stress.

Question 17

1. We are 95% confident that Method A's pass rate is between 0.05 lower and 0.27 higher than Method B's pass rate. (correct answer)
2. Since the interval includes 0, Method B definitely has a higher pass rate than Method A.
3. There is a 95% probability that $p_A$ is between $-0.05$ and 0.27.
4. We are 95% confident that $p_B - p_A$ is between $-0.05$ and 0.27.
5. 95% of all students would have differences in passing between $-0.05$ and 0.27 when switching methods.

Explanation: This question assesses interpreting a confidence interval for pA - pB, the difference in quiz pass rates between review methods. The 95% interval (-0.05, 0.27) means we are 95% confident that Method A's rate is between 0.05 lower and 0.27 higher than Method B's. A distractor like choice D states the interval for pB - pA but fails to correctly invert the bounds. Choice B wrongly infers a definite superiority from including 0. Mini-lesson: construct a CI for p1 - p2 by adding/subtracting z* times the pooled or unpooled standard error from the sample difference; it provides a range of believable differences. With endpoints straddling 0, the data do not provide evidence of a significant difference between methods.

Question 18

1. We are 94% confident that City X's public-transit proportion is between 0.06 lower and 0.09 higher than City Y's public-transit proportion. (correct answer)
2. We are 94% confident that City Y's public-transit proportion is between 0.06 lower and 0.09 higher than City X's public-transit proportion.
3. There is a 94% probability that the interval $(-0.06,\ 0.09)$ contains the sample difference $\hat p_X - \hat p_Y$.
4. Since 0 is in the interval, City X has a lower public-transit proportion than City Y in the population.
5. 94% of commuters in City X use public transit, and City Y differs by between $-0.06$ and 0.09.

Explanation: This question tests interpreting a confidence interval for pX - pY, the difference in public transit use between cities. The 94% interval (-0.06, 0.09) suggests we are 94% confident that City X's proportion is between 0.06 lower and 0.09 higher than City Y's. Choice B distracts by providing incorrect bounds for pY - pX, not properly reflecting the negation. Choice D wrongly concludes a definite difference despite including 0. Mini-lesson: confidence intervals for p1 - p2 use the difference of sample proportions plus/minus a margin of error; they cover the true value in C% of repeated samples. Straddling 0 means the data are consistent with no population difference.

Question 19

1. There is a 90% probability that $p_A - p_B$ is between $-0.16$ and 0.08.
2. We are 90% confident that Offer B's renewal proportion is between 0.16 lower and 0.08 higher than Offer A's renewal proportion.
3. We are 90% confident that Offer A's renewal proportion is between 0.16 lower and 0.08 higher than Offer B's renewal proportion. (correct answer)
4. Because the interval contains 0, Offer A and Offer B have exactly the same renewal proportion in the population.
5. 90% of individual members have renewal probabilities between $-0.16$ and 0.08 when comparing offers.

Explanation: This question evaluates understanding a confidence interval for pA - pB, the difference in membership renewal proportions. The 90% interval (-0.16, 0.08) means we are 90% confident that Offer A's proportion is between 0.16 lower and 0.08 higher than Offer B's. Choice B is a distractor with incorrect bounds for pB - pA, swapping the magnitudes without proper adjustment. Choice A wrongly uses 'probability' for the parameter. Mini-lesson: CIs for p1 - p2 account for sampling variability via the standard error, offering confidence that the true difference is captured. The interval including 0 suggests no significant difference between offers.

Question 20

1. We are 90% confident that County A's flu-shot proportion exceeds County B's by between 0.04 and 0.18. (correct answer)
2. There is a 90% probability that $p_A - p_B$ falls between 0.04 and 0.18.
3. We are 90% confident that $p_B - p_A$ is between 0.04 and 0.18.
4. Because 0 is not in the interval, County B must have a higher flu-shot proportion than County A.
5. About 90% of adults in County A received a flu shot, and the difference from County B is between 0.04 and 0.18.

Explanation: This question evaluates understanding of a confidence interval for the difference of two proportions, here pA - pB for flu-shot rates in two counties. The 90% confidence interval (0.04, 0.18) means we are 90% confident that County A's proportion exceeds County B's by between 0.04 and 0.18. Choice B is a distractor that incorrectly uses 'probability' instead of 'confidence,' misrepresenting the interval as a direct probability on the parameter. Choice C reverses the subtraction order, wrongly suggesting pB exceeds pA. For a mini-lesson: confidence intervals for p1 - p2 estimate the range where the true difference likely falls, with the confidence level reflecting the long-run success rate of the method. The endpoints 0.04 and 0.18, being positive, support that County A has a higher flu-shot proportion without including 0 in the interval.