- A restaurant owner compares the proportion of customers who rate their experience as "excellent" after two different menu designs are implemented on different weeks. From independent random samples of customers, a 95% confidence interval for the difference in excellent-rating proportions (Week 1 menu minus Week 2 menu) is (−0.15,−0.02). Which interpretation is correct?
Answer: We are 95% confident that the Week 1 menu leads to an excellent-rating proportion that is 0.02 to 0.15 lower than the Week 2 menu in the population.
This question tests understanding of negative confidence intervals in a restaurant context. The interval (-0.15, -0.02) for p_Week1 - p_Week2 being entirely negative means Week 1's menu has a lower excellent-rating proportion than Week 2's menu. Choice C incorrectly reverses the order and claims Week 1 is better when the negative interval shows Week 1 is worse. Choice A correctly interprets that Week 1's proportion is 0.02 to 0.15 (or 2 to 15 percentage points) lower than Week 2's. Remember that confidence intervals estimate population parameters, not sample statistics, and that negative values for p₁ - p₂ indicate the first group has the lower proportion in the population.
- A clinic compares the proportion of patients who report side effects for two medications. In a random sample, 22 of 140 patients taking Drug S report side effects and 30 of 150 patients taking Drug T report side effects. A 98% confidence interval for (pS−pT) is (−0.12,0.01). Which interpretation is correct?
Answer: We are 98% confident that the true difference in side-effect proportions (pS−pT) is between −0.12 and 0.01.
This question tests interpretation of a confidence interval containing zero for medication side effects. The interval (-0.12, 0.01) for (pₛ - pₜ) includes both negative and positive values, indicating uncertainty about which drug has more side effects. Choice B correctly states we are 98% confident that the true difference in side-effect proportions is between -0.12 and 0.01. Choice A incorrectly concludes exact equality when the interval only suggests we cannot determine which proportion is larger. Choice C incorrectly assigns probability to parameter equality. Choice D makes an unjustified safety conclusion. Choice E incorrectly refers to sample differences in future samples. When zero is in the interval, we cannot conclude there's a difference between the population proportions. - 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) is (−0.06,0.12). Which interpretation is correct?
Answer: We are 95% confident that the true difference in flu-shot proportions (pX−pY) is between −0.06 and 0.12.
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 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) is (−0.18,−0.02). Which interpretation is correct?
Answer: We are 90% confident that the true difference in purchase proportions (pA−pB) is between −0.18 and −0.02.
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 wildlife biologist compares the proportion of tagged fish recaptured in two lakes using independent random samples of tagged fish. In Lake A, 52 of 160 tagged fish were recaptured; in Lake B, 60 of 150 tagged fish were recaptured. A 90% confidence interval for pA−pB is (−0.17, −0.01). Which interpretation is correct?
Answer: We are 90% confident that the true difference pA−pB is between −0.17 and −0.01, meaning Lake A's recapture proportion is lower than Lake B's by 0.01 to 0.17.
This question involves a negative confidence interval for pA - pB. The interval (-0.17, -0.01) is entirely negative, meaning Lake A has a lower recapture proportion than Lake B. Choice B correctly interprets this: we are 90% confident that pA - pB is between -0.17 and -0.01, meaning Lake A's proportion is lower by 0.01 to 0.17. Choice A incorrectly states Lake A is higher. Choice C treats confidence as probability. Choice D incorrectly concludes equality when 0 is not in the interval. Choice E confuses the difference with a single proportion. - 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−p2 is (0.01, 0.11). Which interpretation is correct?
Answer: We are 95% confident that Version 1's 2FA proportion is 0.01 to 0.11 higher than Version 2's 2FA proportion.
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.
- A sports analyst compares the proportion of free throws made by two players over a season. From random samples of attempts, Player 1 made 85 of 120 and Player 2 made 72 of 115. A 95% confidence interval for (p1−p2) is (−0.01,0.17). Which interpretation is correct?
Answer: We are 95% confident that the true difference in free-throw proportions (p1−p2) is between −0.01 and 0.17.
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. - A sports scientist compares the proportion of athletes who report improved performance after using two training plans. From independent random samples, a 90% confidence interval for the difference in improvement proportions (Plan A minus Plan B) is (−0.03,0.09). Which interpretation is correct?
Answer: We are 90% confident that the true difference pA−pB is between −0.03 and 0.09.
This question examines interpretation of a confidence interval that spans both negative and positive values. The interval (-0.03, 0.09) for pA - pB includes zero, meaning Plan A could have a slightly lower rate (up to 3 percentage points), the same rate, or a higher rate (up to 9 percentage points) than Plan B. Choice B incorrectly concludes Plan A is definitely worse just because the interval includes negative values. Choice A correctly interprets the interval as our 90% confidence about the true population difference. When a confidence interval includes both negative and positive values, we cannot determine which population has the higher proportion—the data is consistent with either possibility or with no difference at all. - A gym compared the proportion of members who renew after a trial month for two membership offers. For Offer A, 54 of 120 trial members renewed; for Offer B, 63 of 130 trial members renewed. A 90% confidence interval for pA−pB is (−0.16, 0.08). Which interpretation is correct?
Answer: 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.
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.
- A company tested two website designs to see which leads to more purchases. Among 500 visitors shown Design 1, 62 made a purchase; among 520 visitors shown Design 2, 78 made a purchase. A 99% confidence interval for the difference in purchase rates, p1−p2, is (−0.05, 0.01). Which interpretation is correct?
Answer: 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.
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.
- 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) is (0.02,0.20). Which interpretation is correct?
Answer: We are 95% confident that the true difference in proportions (p9−p10) is between 0.02 and 0.20.
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 university compares the proportion of students who graduate in 4 years for two programs. In a random sample, 140 of 200 students in Program A graduate in 4 years and 118 of 190 students in Program B graduate in 4 years. A 92% confidence interval for (pA−pB) is (0.01,0.17). Which interpretation is correct?
Answer: We are 92% confident that the true difference in 4-year graduation proportions (pA−pB) is between 0.01 and 0.17.
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. - 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−pS is (0.01, 0.16). Which interpretation is correct?
Answer: We are 98% confident that the North's support proportion is between 0.01 and 0.16 higher than the South's support proportion.
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 school compared the proportion of students who prefer online homework in two grades. In a random sample, 84 of 150 ninth-graders and 72 of 160 tenth-graders said they prefer online homework. A 95% confidence interval for the difference in population proportions, p9−p10, is (0.02, 0.20). Which interpretation is correct?
Answer: 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.
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 (p^1−p^2) \pm z^* \sqrt{\frac{\hat{p}_1(1−p^1)}{n_1} + \frac{\hat{p}_2(1−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. - A school compares the proportion of students who prefer online homework between two grade levels. A random sample of 180 ninth-graders and 200 tenth-graders was taken, and a 95% confidence interval for the difference in proportions (ninth minus tenth) was computed as (0.04,0.18). Which interpretation is correct?
Answer: We are 95% confident that the true difference in population proportions pninth−ptenth is between 0.04 and 0.18.
This question tests understanding of confidence intervals for the difference between two population proportions. The interval (0.04, 0.18) estimates the true difference p_ninth - p_tenth, where positive values indicate ninth-graders have a higher preference rate. Choice B incorrectly interprets this as a single proportion for ninth-graders only, missing that it's about the difference. Choice D correctly states we are 95% confident the true population difference falls within this interval. When interpreting confidence intervals for differences, remember that the interval estimates the parameter (population difference), not the statistic (sample difference), and that confidence refers to the method's long-run success rate, not probability for a specific interval. - A teacher compared the proportion of students who pass a quiz after two different review methods. In one class using Method A, 31 of 50 students passed; in another class using Method B, 28 of 55 students passed. A 95% confidence interval for pA−pB is (−0.05, 0.27). Which interpretation is correct?
Answer: 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.
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.
- A city compared the proportion of households that recycle regularly in two neighborhoods. In random samples, 98 of 140 households in Neighborhood East and 84 of 150 households in Neighborhood West reported recycling regularly. A 92% confidence interval for pE−pW is (0.03, 0.25). Which interpretation is correct?
Answer: We are 92% confident that Neighborhood East's recycling proportion is between 0.03 and 0.25 higher than Neighborhood West's recycling proportion.
This question examines the interpretation of a confidence interval for pE - pW, the difference in recycling proportions between neighborhoods. The 92% interval (0.03, 0.25) indicates we are 92% confident that East's proportion is between 0.03 and 0.25 higher than West's. Choice A distracts by reversing the order to pW - pE without flipping the interval signs. Choice E mistakenly applies the interval to sample proportions instead of population ones. In a mini-lesson: CIs for p1 - p2 are built around the observed difference with a margin of error based on sample sizes and variability, capturing the true difference with C% confidence. The positive endpoints here, excluding 0, suggest East has a significantly higher recycling rate.
- 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−p2 is (0.02, 0.30). Which interpretation is correct?
Answer: We are 98% confident that the true difference in passing rates, p1−p2, is between 0.02 and 0.30.
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 university compares the proportion of students who pass a placement test after using two different prep programs. From independent random samples, a 92% confidence interval for the difference in pass rates (Program A minus Program B) is (0.00,0.14). Which interpretation is correct?
Answer: We are 92% confident that pA−pB is between 0.00 and 0.14.
This question tests understanding of confidence intervals that include the boundary value 0.00. The interval (0.00, 0.14) for pA - pB suggests Program A's pass rate could be equal to or up to 14 percentage points higher than Program B's. Choice C incorrectly claims that including 0.00 means the programs have identical pass rates; it only means we cannot rule out equality. Choice B correctly interprets the interval as our confidence about the true population difference. When a confidence interval's lower bound is exactly 0, we're on the borderline of statistical significance—we have some evidence of a difference but cannot completely rule out equality at the given confidence level. - A researcher compares the proportion of patients who report side effects for two medications using independent random samples. For Medication X, 28 of 80 patients report side effects; for Medication Y, 18 of 75 patients report side effects. A 92% confidence interval for pX−pY is (0.02, 0.24). Which interpretation is correct?
Answer: We are 92% confident that Medication X's side-effect proportion is between 0.02 and 0.24 higher than Medication Y's side-effect proportion.
This question tests interpretation of a confidence interval for the difference in side-effect proportions. The interval (0.02, 0.24) for pX - pY is entirely positive, meaning Medication X has a higher side-effect proportion than Medication Y. Choice B correctly interprets this: we are 92% confident that Medication X's side-effect proportion is between 0.02 and 0.24 higher than Medication Y's. Choice A only describes one proportion. Choice C misinterprets the positive interval. Choice D incorrectly treats confidence as probability for the samples. Choice E reverses the conclusion (X has more side effects, not Y).
