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.

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 $(0.04, 0.20)$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $(p_1 - p_2)$.

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.

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.

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.

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 $(0.03,\ 0.21)$.

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.

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.

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.

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.