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AP Statistics

AP Statistics Help: Confidence Intervals Difference Of Two Proportions

Review real example questions for Confidence Intervals Difference Of Two Proportions in AP Statistics.

Question 1

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 p1−p2p_1 - p_2p1​−p2​ is (0.01, 0.11)(0.01,\ 0.11)(0.01, 0.11). Which interpretation is correct?

  1. There is a 95% chance that the difference p1−p2p_1 - p_2p1​−p2​ 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.
  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

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 p1−p2p_1 - p_2p1​−p2​ is (0.04, 0.20)(0.04,\ 0.20)(0.04, 0.20). Which interpretation is correct?

  1. We are 95% confident that p2−p1p_2 - p_1p2​−p1​ is between 0.04 and 0.20.
  2. If we repeated the sampling many times, 95% of the intervals would contain the sample difference p^1−p^2\hat p_1 - \hat p_2p^​1​−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.
  5. There is a 95% probability that the true difference p1−p2p_1 - p_2p1​−p2​ is outside the interval (0.04,0.20)(0.04, 0.20)(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

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 p1−p2p_1 - p_2p1​−p2​ is (0.01, 0.11)(0.01,\ 0.11)(0.01, 0.11). Which interpretation is correct?

  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.
  2. There is a 95% probability that the interval (0.01,0.11)(0.01, 0.11)(0.01,0.11) will contain p^1−p^2\hat p_1 - \hat p_2p^​1​−p^​2​.
  3. Because 0 is not in the interval, we know for sure that p1−p2=0.06p_1 - p_2 = 0.06p1​−p2​=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

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 p1−p2p_1 - p_2p1​−p2​ is (0.02, 0.30)(0.02,\ 0.30)(0.02, 0.30). Which interpretation is correct?

  1. We are 98% confident that the true difference in passing rates, p1−p2p_1 - p_2p1​−p2​, is between 0.02 and 0.30.
  2. There is a 98% chance that Session 1 will cause a student to pass the exam.
  3. We are 98% confident that p2−p1p_2 - p_1p2​−p1​ 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

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 pR−pHp_R - p_HpR​−pH​ is (−0.18, −0.04)(-0.18,\ -0.04)(−0.18, −0.04). Which interpretation is correct?

  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.
  2. There is a 90% probability that pR−pHp_R - p_HpR​−pH​ equals a value between −0.18-0.18−0.18 and −0.04-0.04−0.04.
  3. Because the interval is negative, 90% of renters and 90% of homeowners support the fee.
  4. We are 90% confident that pH−pRp_H - p_RpH​−pR​ is between −0.18-0.18−0.18 and −0.04-0.04−0.04.
  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

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 pN−pSp_N - p_SpN​−pS​ is (0.01, 0.16)(0.01,\ 0.16)(0.01, 0.16). Which interpretation is correct?

  1. There is a 98% chance that the true proportions pNp_NpN​ and pSp_SpS​ 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.
  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 p^N−p^S\hat p_N - \hat p_Sp^​N​−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

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 (pX−pY)(p_X - p_Y)(pX​−pY​) is (−0.06,0.12)(-0.06, 0.12)(−0.06,0.12). Which interpretation is correct?

  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 (pX−pY)(p_X - p_Y)(pX​−pY​) is between −0.06-0.06−0.06 and 0.120.120.12.
  3. There is a 95% probability that the true difference (pX−pY)(p_X - p_Y)(pX​−pY​) 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 −0.06-0.06−0.06 and 0.120.120.12 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

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 (p9−p10)(p_9 - p_{10})(p9​−p10​) is (0.02,0.20)(0.02, 0.20)(0.02,0.20). Which interpretation is correct?

  1. There is a 95% probability that the true difference (p9−p10)(p_9 - p_{10})(p9​−p10​) is between 0.02 and 0.20.
  2. We are 95% confident that the true difference in proportions (p9−p10)(p_9 - p_{10})(p9​−p10​) is between 0.02 and 0.20.
  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 p9p_9p9​ and p10p_{10}p10​.
  5. We are 95% confident that the difference in sample proportions (p^9−p^10)(\hat p_9 - \hat p_{10})(p^​9​−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

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 (pA−pB)(p_A - p_B)(pA​−pB​) is (−0.18,−0.02)(-0.18, -0.02)(−0.18,−0.02). Which interpretation is correct?

  1. We are 90% confident that Design A’s purchase proportion is between −0.18-0.18−0.18 and −0.02-0.02−0.02.
  2. Because the interval is negative, we are 90% confident that pBp_BpB​ is between 0.02 and 0.18 greater than pAp_ApA​.
  3. There is a 90% chance that (pA−pB)(p_A - p_B)(pA​−pB​) is negative for this experiment.
  4. We are 90% confident that the true difference in purchase proportions (pA−pB)(p_A - p_B)(pA​−pB​) is between −0.18-0.18−0.18 and −0.02-0.02−0.02.
  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

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 (p1−p2)(p_1 - p_2)(p1​−p2​) is (−0.05,0.13)(-0.05, 0.13)(−0.05,0.13). Which interpretation is correct?

  1. We are 99% confident that the true difference in support proportions (p1−p2)(p_1 - p_2)(p1​−p2​) is between −0.05-0.05−0.05 and 0.130.130.13.
  2. There is a 99% probability that p1=p2p_1 = p_2p1​=p2​ 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 (p^1−p^2)(\hat p_1 - \hat p_2)(p^​1​−p^​2​) is between −0.05-0.05−0.05 and 0.130.130.13 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.