Confidence Intervals: Difference of Two Proportions
Help Questions
AP Statistics › Confidence Intervals: Difference of Two Proportions
A tech company compares the proportion of users who enable two-factor authentication (2FA) on two app versions. In independent random samples, 410 of 800 users on Version 1 enabled 2FA and 372 of 820 users on Version 2 enabled 2FA. A 95% confidence interval for $p_1 - p_2$ is $(0.01,\ 0.11)$. Which interpretation is correct?
There is a 95% chance that the difference $p_1 - p_2$ is between 0.01 and 0.11.
Because 0 is not in the interval, the sample proportions must be equal.
We are 95% confident that the proportion of users who enable 2FA is between 0.01 and 0.11 for Version 1.
We are 95% confident that Version 2’s 2FA proportion is 0.01 to 0.11 higher than Version 1’s 2FA proportion.
We are 95% confident that Version 1’s 2FA proportion is 0.01 to 0.11 higher than Version 2’s 2FA proportion.
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.
Two independent random samples are taken to compare the proportion of adults who drink coffee daily in two regions. In Region 1, 156 of 260 adults drink coffee daily; in Region 2, 120 of 250 adults drink coffee daily. A 95% confidence interval for $p_1 - p_2$ is $(0.04,\ 0.20)$. Which interpretation is correct?
There is a 95% probability that the true difference $p_1 - p_2$ is outside the interval $(0.04, 0.20)$.
We are 95% confident that $p_2 - p_1$ is between 0.04 and 0.20.
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.
We are 95% confident that the proportion of adults who drink coffee daily is between 0.04 and 0.20 in Region 1.
If we repeated the sampling many times, 95% of the intervals would contain the sample difference $\hat p_1 - \hat p_2$.
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.
Two independent random samples are used to compare the proportion of voters who approve of Candidate A in two counties. In County 1, 310 of 500 approve; in County 2, 295 of 520 approve. A 95% confidence interval for $p_1 - p_2$ is $(0.01,\ 0.11)$. Which interpretation is correct?
We are 95% confident that Candidate A’s approval proportion in County 1 is 0.01 to 0.11 higher than in County 2.
There is a 95% probability that the interval $(0.01, 0.11)$ will contain $\hat p_1 - \hat p_2$.
We are 95% confident that Candidate A’s approval proportion in County 2 is 0.01 to 0.11 higher than in County 1.
We are 95% confident that between 1% and 11% of all voters approve of Candidate A in County 1.
Because 0 is not in the interval, we know for sure that $p_1 - p_2 = 0.06$.
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.
A university compares the proportion of students who pass an exam after two different review sessions. In independent random samples, 45 of 60 students who attended Session 1 passed and 38 of 62 students who attended Session 2 passed. A 98% confidence interval for $p_1 - p_2$ is $(0.02,\ 0.30)$. Which interpretation is correct?
We are 98% confident that the true difference in passing rates, $p_1 - p_2$, is between 0.02 and 0.30.
We are 98% confident that $p_2 - p_1$ is between 0.02 and 0.30.
Because the interval does not include 0, there is no difference between the two sessions in the population.
There is a 98% chance that Session 1 will cause a student to pass the exam.
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.
A city surveys two independent random samples to compare the proportion who support a new recycling fee. Among 210 renters, 98 support the fee; among 190 homeowners, 105 support the fee. A 90% confidence interval for $p_R - p_H$ is $(-0.18,\ -0.04)$. Which interpretation is correct?
We are 90% confident that $p_H - p_R$ is between $-0.18$ and $-0.04$.
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.
There is a 90% probability that $p_R - p_H$ equals a value between $-0.18$ and $-0.04$.
Since 0 is not in the interval, there is no difference between renters and homeowners in the population.
Because the interval is negative, 90% of renters and 90% of homeowners support the fee.
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.
A political scientist compared the proportion of voters who support a ballot measure in two regions. In random samples, 210 of 350 voters in the North region and 188 of 360 voters in the South region supported the measure. A 98% confidence interval for $p_N - p_S$ is $(0.01,\ 0.16)$. Which interpretation is correct?
We are 98% confident that the North’s support proportion is between 0.01 and 0.16 higher than the South’s support proportion.
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.
Because 0 is not in the interval, the probability that a randomly selected voter supports the measure is between 0.01 and 0.16.
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.
We are 98% confident that the South’s support proportion is between 0.01 and 0.16 higher than the North’s support proportion.
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.
A public health study compares the proportion of adults who received a flu shot in two counties. In County X, 156 of 260 adults in a random sample received a flu shot; in County Y, 170 of 300 adults in a random sample received a flu shot. A 95% confidence interval for $(p_X - p_Y)$ is $(-0.06, 0.12)$. Which interpretation is correct?
There is a 95% probability that the true difference $(p_X - p_Y)$ is exactly 0.
If we took many samples, 95% of adults would fall within $-0.06$ and $0.12$ of being vaccinated.
Because 0 is in the interval, we are 95% confident that County X has a lower flu-shot proportion than County Y.
We are 95% confident that 6% to 12% of adults in County X received a flu shot.
We are 95% confident that the true difference in flu-shot proportions $(p_X - p_Y)$ is between $-0.06$ and $0.12$.
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.
A school compares the proportion of students who prefer online homework between two grades. In a random sample, 78 of 120 ninth-graders and 60 of 110 tenth-graders said they prefer online homework. A 95% confidence interval for the difference in population proportions $(p_9 - p_{10})$ is $(0.02, 0.20)$. Which interpretation is correct?
There is a 95% probability that the true difference $(p_9 - p_{10})$ is between 0.02 and 0.20.
Because 0 is not in the interval, there is no difference between $p_9$ and $p_{10}$.
We are 95% confident that the true difference in proportions $(p_9 - p_{10})$ is between 0.02 and 0.20.
About 95% of ninth-graders prefer online homework, and about 95% of tenth-graders do too.
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.
A company tests two website designs. Among 200 randomly selected visitors shown Design A, 54 made a purchase; among 180 randomly selected visitors shown Design B, 63 made a purchase. A 90% confidence interval for $(p_A - p_B)$ is $(-0.18, -0.02)$. Which interpretation is correct?
Since 0 is not in the interval, the two sample proportions must be equal.
Because the interval is negative, we are 90% confident that $p_B$ is between 0.02 and 0.18 greater than $p_A$.
We are 90% confident that the true difference in purchase proportions $(p_A - p_B)$ is between $-0.18$ and $-0.02$.
There is a 90% chance that $(p_A - p_B)$ is negative for this experiment.
We are 90% confident that Design A’s purchase proportion is between $-0.18$ and $-0.02$.
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.
A city surveys two neighborhoods about support for a new park. In a random sample, 96 of 160 residents in Neighborhood 1 support the park and 84 of 150 residents in Neighborhood 2 support the park. A 99% confidence interval for $(p_1 - p_2)$ is $(-0.05, 0.13)$. Which interpretation is correct?
Because 0 is in the interval, we are 99% confident that Neighborhood 1 has a higher support proportion than Neighborhood 2.
There is a 99% probability that $p_1 = p_2$ because 0 is in the interval.
We are 99% confident that the true difference in support proportions $(p_1 - p_2)$ is between $-0.05$ and $0.13$.
About 99% of all residents in both neighborhoods support the park.
We are 99% confident that the sample difference $(\hat p_1 - \hat p_2)$ is between $-0.05$ and $0.13$ for all samples.
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.