25 hard SAT Inference from samples & margin of error questions
Real questions from the SAT Climb bank, all at the hard difficulty tier. Pick an answer before you open the explanation. Every question tells you why the answer is right and why each wrong choice is tempting.
Math · Problem-Solving and Data Analysis~1 per testHard tier
Reads the estimate as exact instead of an interval.
Generalizes beyond the population that was actually sampled.
Thinks a bigger sample widens (rather than narrows) the margin.
Question 1Hard
In a survey of 900 randomly selected adults, 63% reported exercising at least three times per week. The margin of error for this survey is±4 percentage points. A researcher claims that more than 65% of all adults in the population exercise at least three times per week. Based on the survey results, which of the following is true?
Show the answer and explanation
Why B is right
The confidence interval is 63%±4%, or 59% to 67%. Since this interval includes values both above and below 65%, the survey does not provide strong evidence for the claim that more than 65% of adults exercise regularly.
Why the others are wrong
AThis ignores the margin of error entirely, comparing only the point estimate to the claimed value.
CThis treats the margin of error as if it were itself evidence for the claim, rather than recognizing it creates uncertainty around the estimate.
DA sample size of 900 is quite large for survey purposes; this incorrectly suggests the sample size makes the result unreliable.
Question 2Hard
A researcher surveyed a random sample of 240 customers at a store during one week. Of those surveyed, 156 customers purchased items using a mobile payment app. The store had 4,800 customers during that week. If the sample is representative of all customers that week, and the researcher expects the same proportion to continue over the next 6 months, what is the best estimate of the total number of customers who will use a mobile payment app over the next 6 months if the store is projected to have 28,800 customers during that period?
Show the answer and explanation
Why B is right
The sample proportion is 156/240 = 0.65. To estimate the number of customers who will use mobile payment over the next 6 months, multiply this proportion by the projected customer count: 0.65 × 28,800 = 18,720.
Why the others are wrong
AThis incorrectly multiplies the sample count of mobile users (156) by the ratio of future to current customers (28,800/4,800 = 6), giving 156 × 20 = 3,120, but this fails to account for the proper proportion scaling.
CThis uses the number of customers who did NOT use mobile payment in the sample (240 - 156 = 84) and incorrectly scales: 84 × (28,800/240) = 10,080, then further miscalculates.
DThis correctly finds the proportion (0.65) but only multiplies by the single-week customer count scaled incorrectly: 0.65 × 4,800 × 4 = 12,480, rather than using the full 6-month projection.
Question 3Hard
A retail manager surveyed 175 customers who made purchases using the self-checkout lanes on a Sunday afternoon between 2:00 PM and 4:00 PM about their checkout experience. What is the largest population to which the results of this survey can be generalized?
Show the answer and explanation
Why C is right
This is a convenience sample limited to self-checkout users on a specific day (Sunday) during a specific two-hour window (2:00-4:00 PM). The results can only be generalized to customers who match all three criteria: self-checkout, Sunday, and that specific afternoon time period.
Why the others are wrong
AThis overgeneralizes by assuming the convenience sample represents all store customers, when many may use staffed lanes, shop at different times, or shop on weekdays instead of Sundays.
BThis overgeneralizes by extending the specific 2:00-4:00 PM window to the entire Sunday afternoon, when morning or evening self-checkout users may have different experiences or preferences.
DThis overgeneralizes by ignoring that the sample was taken only on Sunday afternoons during a specific window, so it cannot represent self-checkout users at other times or days.
Question 4Hard
A political poll surveyed 2500 randomly selected likely voters and found that 58% plan to vote for Candidate A, with a margin of error of±2 percentage points at a 95% confidence level. A second poll surveyed n randomly selected likely voters from the same population and found that 54% plan to vote for Candidate A, with a margin of error of±4 percentage points at the same confidence level. Which of the following best describes the relationship between the two polls?
Show the answer and explanation
Why A is right
The second poll has a margin of error (±4 percentage points) that is double that of the first poll (±2 percentage points). Since margin of error is inversely proportional to the square root of sample size, doubling the margin means dividing the sample size by 4. Therefore, n=2500÷4=625.
Why the others are wrong
BThis incorrectly reverses the relationship, assuming that a larger margin of error requires a larger sample size. In fact, larger samples produce smaller margins of error.
CThis conflates the confidence intervals with a comparison of support levels. The first poll's CI is (56%, 60%) and the second's is (50%, 58%), which do overlap at (56%, 58%). Moreover, the question asks about the relationship between polls, not whether support changed.
DWhile it's true the difference is 2 percentage points, this statement ignores the ± notation and doesn't address the sample size relationship that the question is testing.
Question 5Hard
A polling organization surveyed 1600 randomly selected registered voters and found that 51% plan to vote for Candidate X, with a margin of error of±2.5 percentage points. A second independent poll of 400 randomly selected registered voters from the same population found that 47% plan to vote for Candidate X. Assuming the margin of error is inversely proportional to the square root of the sample size, do the confidence intervals from these two polls overlap?
Show the answer and explanation
Why C is right
For the second poll, the sample size is one-fourth that of the first, so its margin of error is 2.5%×2=±5 percentage points. The second poll's CI is 47%±5% = 42% to 52%. The first poll's CI is 51%±2.5% = 48.5% to 53.5%. These intervals overlap from 48.5% to 52%.
Why the others are wrong
AThis ignores the margins of error entirely and only compares the point estimates, which is insufficient for determining whether confidence intervals overlap.
BThis correctly identifies the second poll's CI but incorrectly states the intervals do not overlap; they do overlap in the range 48.5% to 52%.
DWhile the statement about sample size and margin of error is mathematically correct, it does not address whether the confidence intervals overlap.
Question 6Hard
A polling firm surveyed a random sample of residents in a county to estimate support for a new park development. The survey found that 58% of the sample supported the project. At a 95% confidence level, the margin of error was calculated to be±4 percentage points. The firm wants to reduce the margin of error to±2 percentage points at the same confidence level for a follow-up survey. By what factor must the sample size be increased?
Show the answer and explanation
Why B is right
The margin of error is inversely proportional to the square root of the sample size. To reduce the margin of error by a factor of 2 (from 4 to 2 percentage points), the sample size must be increased by a factor of 22=4.
Why the others are wrong
AThis incorrectly applies the reduction factor directly instead of squaring it, failing to account for the square root relationship between sample size and margin of error.
CThis incorrectly cubes the reduction factor or applies a wrong exponential relationship, yielding 2³ = 8 instead of 2² = 4.
DThis falsely claims the margin of error is fixed for a given confidence level, ignoring that increasing sample size reduces the margin of error at any fixed confidence level.
Question 7Hard
A survey of 900 randomly selected adults found that 63% prefer online shopping to in-store shopping, with a margin of error of±3.2 percentage points. Based on this information, which of the following best describes the 95% confidence interval for the proportion of all adults who prefer online shopping?
Show the answer and explanation
Why A is right
The confidence interval is constructed by taking the sample proportion and adding/subtracting the margin of error: 63%±3.2% gives the interval from 59.8% to 66.2%.
Why the others are wrong
BThis incorrectly uses only the lower bound with the margin of error, omitting the symmetric upper bound.
CThis incorrectly applies the margin of error in only one direction, giving only the upper portion of the interval.
DThis incorrectly doubles the margin of error, confusing it with a different confidence level or calculation.
Question 8Hard
In a study, 180 employees at a company were randomly selected and surveyed about their remote work preferences. Of those surveyed, 117 employees preferred remote work at least 3 days per week. The company currently has 2,700 employees. If the sample accurately represents current employee preferences and the company plans to hire additional employees over the next 6 months so that the total workforce will be 3,600 employees, what is the best estimate of the number of employees who will prefer remote work at least 3 days per week at that time?
Show the answer and explanation
Why C is right
The sample proportion is 117/180 = 0.65. Applying this proportion to the projected workforce of 3,600 employees gives 0.65 × 3,600 = 2,340 employees who will prefer remote work at least 3 days per week.
Why the others are wrong
AThis incorrectly applies the proportion to the current workforce only: 0.65 × 2,700 = 1,755, failing to use the projected 6-month workforce size.
BThis multiplies the sample count (117) by the ratio of new to current workforce (900/2,700 = 1/3), giving 117 × 5 = 585, which does not properly scale the proportion to the full future population.
DThis uses the complement proportion (63/180 = 0.35 for those NOT preferring remote work) and incorrectly computes 0.65 × 2,700 + 0.35 × 900 = 2,106.
Question 9Hard
A hospital administrator surveyed 250 patients who were in the emergency room waiting area on a Saturday night about their satisfaction with wait times. What is the largest population to which the results of this survey can be generalized?
Show the answer and explanation
Why A is right
This is a convenience sample of patients who happened to be in the emergency room waiting area at a specific time. The results can only be generalized to patients in the emergency room on Saturday nights, not to all emergency room patients or all hospital patients.
Why the others are wrong
BThis overgeneralizes by ignoring that Saturday night emergency room visits may differ significantly from visits on weekday mornings or other times in terms of case types and volumes.
CThis overgeneralizes by assuming emergency room patients on Saturday nights represent all hospital patients, including those in scheduled surgeries, routine appointments, or other departments.
DThis overgeneralizes by extending Saturday night to all weekend times, when Sunday morning emergency room patients may have different characteristics than Saturday night patients.
Question 10Hard
A university conducted two independent surveys about student satisfaction. The first survey sampled 576 randomly selected students and reported a margin of error of±4 percentage points at a 95% confidence level. The second survey, using identical methodology, sampled 2304 randomly selected students from the same population. What is the approximate margin of error for the second survey at the same confidence level?
Show the answer and explanation
Why A is right
The second survey has a sample size of 2304, which is 4 times the first survey's 576. Since margin of error is inversely proportional to the square root of sample size, increasing the sample size by a factor of 4 decreases the margin of error by a factor of 4=2. Therefore, the margin of error is 4%÷2=±2 percentage points.
Why the others are wrong
BThis incorrectly assumes margin of error increases with sample size. In reality, larger samples provide more precise estimates with smaller margins of error.
CThis drops the ± symbol and also uses an incorrect value. While ±1 percentage point would result from a sample size 16 times larger than 576, the actual increase is only 4 times.
DThis confuses the margin of error with confidence interval width. The margin of error is ±2 percentage points, which produces a confidence interval width of 4 percentage points (from point estimate - 2% to point estimate + 2%).
Question 11Hard
A pollster surveys 900 randomly selected likely voters and finds that 46% plan to vote for Measure Z, with a margin of error of±3.3 percentage points. A second pollster surveys 3,600 randomly selected likely voters from the same population and finds that 51% plan to vote for Measure Z. Assuming both polls use the same confidence level, which of the following best describes the relationship between the two polls?
Show the answer and explanation
Why A is right
The second sample size (3,600) is 4 times the first (900), so its margin of error is reduced by a factor of 2 to ±1.65 percentage points. The first poll's CI is 42.7% to 49.3%, and the second is 49.35% to 52.65%, which overlap slightly, indicating no statistically significant difference.
Why the others are wrong
BThis incorrectly assumes the margin of error increases when sample size increases, and the calculated margin would make the intervals overlap significantly, contradicting the claim.
CThis assumes the margin of error remains unchanged despite a quadrupling of sample size, which ignores the inverse square root relationship.
DThis applies a factor of 4 instead of 2, and the intervals would actually overlap at the lower bound of the second poll's CI, contradicting the claim of no overlap.
Question 12Hard
A market research firm surveyed 3200 randomly selected households and found that 67% subscribe to streaming services, with a margin of error of±1.7 percentage points. If the firm wants to achieve a margin of error of±0.85 percentage points at the same confidence level, how many households should be surveyed?
Show the answer and explanation
Why D is right
To halve the margin of error from ±1.7 to ±0.85 percentage points, the sample size must be quadrupled. The new sample size is 3200×4=12800 households.
Why the others are wrong
AThis incorrectly assumes that doubling the sample size halves the margin of error.
BThis reverses the relationship, decreasing the sample size when it should be increased.
CThis incorrectly triples the sample size, confusing the relationship between margin of error and sample size.
Question 13Hard
A health clinic surveyed a random sample of 400 patients who visited during a month. Of those surveyed, 92 patients reported using the clinic's online appointment system. The clinic had 8,000 patient visits during that month. If the usage pattern continues and the clinic expects to have 14,000 patient visits over the next 6 months due to expanded services, what is the best estimate of the number of patients who will use the online appointment system during that period?
Show the answer and explanation
Why D is right
The sample proportion is 92/400 = 0.23. Applying this proportion to the projected 14,000 patient visits gives 0.23 × 14,000 = 3,220 patients who will use the online appointment system.
Why the others are wrong
AThis incorrectly applies the proportion to the current month's visits only: 0.23 × 8,000 = 1,840, failing to use the 6-month projection of 14,000 visits.
BThis multiplies the sample count (92) by the visit increase ratio (14,000/8,000 = 1.75) to get approximately 161, which represents a computational error and wrong methodology.
CThis uses the proportion of patients NOT using the system (308/400 = 0.77) and computes 0.77 × 7,000 = 5,390, using both the wrong statistic and wrong base.
Question 14Hard
A university dining services director surveyed 220 students who were eating lunch in the main dining hall on a Wednesday between 12:00 PM and 1:00 PM about their meal preferences. What is the largest population to which the results of this survey can be generalized?
Show the answer and explanation
Why D is right
This is a convenience sample taken at one specific location, on one specific day of the week, during one specific time window. The results can only be generalized to students who match all three constraints: main dining hall, Wednesday, and the 12:00-1:00 PM lunch hour.
Why the others are wrong
AThis overgeneralizes by ignoring that the sample was taken only on Wednesday during a specific one-hour window, so it cannot represent students who eat there at other times or days.
BThis overgeneralizes by assuming the convenience sample represents all university students, when many students may eat elsewhere, at different times, or skip lunch entirely.
CThis overgeneralizes by ignoring that the sample was taken only from the main dining hall on Wednesdays, not from students who eat at that time in other locations or on other days.
Question 15Hard
A health organization surveyed 1600 randomly selected adults and found that 42% exercise regularly, with a margin of error of±2.5 percentage points for a 95% confidence level. A follow-up survey of 400 randomly selected adults from the same population found that 38% exercise regularly. Assuming the same confidence level and similar sampling methods, what is the approximate margin of error for the follow-up survey?
Show the answer and explanation
Why D is right
The follow-up survey has a sample size that is 1/4 of the original (400 vs 1600). Since margin of error is inversely proportional to the square root of sample size, reducing the sample size by a factor of 4 increases the margin of error by a factor of 4=2. Therefore, the margin of error is 2.5%×2=±5 percentage points.
Why the others are wrong
AThis incorrectly assumes the margin of error decreases when sample size decreases. In fact, smaller samples produce less precise estimates with larger margins of error.
BThis presents a confidence interval rather than the margin of error. While (33%, 43%) would be the approximate 95% CI using a ±5 percentage point margin, the question asks specifically for the margin of error.
CThis omits the ±, treating margin of error as a single-direction adjustment rather than a symmetric range around the point estimate.
Question 16Hard
Two surveys were conducted to estimate support for a new ordinance. Survey A of 400 randomly selected residents found 55% support with a margin of error of±5 percentage points. Survey B of 1,600 randomly selected residents found 48% support. If Survey B has the same confidence level as Survey A, which of the following conclusions is supported?
Show the answer and explanation
Why A is right
Survey B's sample size is 4 times Survey A's, so its margin of error is ±5/2=±2.5 percentage points. Survey A's CI is 50% to 60%, and Survey B's is 45.5% to 50.5%. These do not overlap, indicating a statistically significant difference.
Why the others are wrong
BThis incorrectly assumes margin of error increases with sample size, and a ±10 percentage point margin would create a CI of 38% to 58%, which would overlap substantially with Survey A's CI.
CThis assumes the margin of error stays the same despite a quadrupling of sample size, which would create a CI of 43% to 53% for Survey B, overlapping with Survey A's 50% to 60%.
DThis overstates the reduction factor and the calculated intervals (Survey B: 46.75% to 49.25%) would not overlap with Survey A (50% to 60%), but the overlap calculation is incorrect.
Question 17Hard
A transportation official surveyed 180 commuters who were boarding express trains departing from the city center during the 8:00 AM hour on weekday mornings about their commuting preferences. What is the largest population to which the results of this survey can be generalized?
Show the answer and explanation
Why A is right
This is a convenience sample restricted to a specific train type (express), specific location (city center), specific time (8:00 AM hour), and specific days (weekdays). The results can only be generalized to commuters who meet all these criteria simultaneously.
Why the others are wrong
BThis overgeneralizes by assuming the convenience sample represents all train users, including those on local trains, different stations, different times, or weekend travelers.
CThis overgeneralizes by ignoring that the sample only includes the 8:00 AM hour on weekdays, so it cannot represent express train users at other times or on weekends.
DThis overgeneralizes by ignoring that the sample only includes express train boarders from the city center, not commuters using buses, local trains, or other stations during that hour.
Question 18Hard
An environmental survey of 1800 randomly selected residents found that 67% support a new recycling initiative, with a margin of error of±3 percentage points for a 95% confidence level. If a second survey is conducted with 450 randomly selected residents from the same population using the same methodology, what would be the approximate margin of error for the second survey at the same confidence level?
Show the answer and explanation
Why B is right
The second survey has a sample size of 450, which is 1/4 of the original 1800. When sample size decreases by a factor of 4, the margin of error increases by a factor of 4=2. Therefore, the margin of error for the second survey is 3%×2=±6 percentage points.
Why the others are wrong
AThis incorrectly assumes that reducing sample size by a factor of 4 reduces the margin of error by the same factor. The relationship involves the square root, and smaller samples have larger margins of error, not smaller.
CThis presents a confidence interval rather than answering the question about margin of error. Additionally, the interval width shown (6 percentage points) corresponds to the margin being ±3 percentage points, not the correct ±6.
DThis drops the ± notation and also uses the wrong value. The margin of error for the smaller sample would be ±6 percentage points, not 3.
Question 19Hard
A researcher conducts a survey of 1,200 randomly selected households and finds that 45% have solar panels installed, with a margin of error of±4 percentage points. The researcher wants to reduce the margin of error to±2 percentage points. How many households should be surveyed?
Show the answer and explanation
Why C is right
To reduce the margin of error by half (from ±4 to ±2 percentage points), the sample size must be multiplied by 4, since margin of error is inversely proportional to the square root of sample size. Therefore, 1,200×4=4,800 households.
Why the others are wrong
AThis only doubles the sample size, which would reduce the margin of error by a factor of the square root of 2 (approximately 1.4), not by half.
BThis triples the sample size, which would reduce the margin of error by a factor of the square root of 3 (approximately 1.7), not by half.
DThis multiplies the sample size by 8, which would reduce the margin of error by a factor of the square root of 8 (approximately 2.8), overshooting the target.
Question 20Hard
A fitness instructor surveyed 150 members who attended her 6:00 AM spin class on Monday morning about their exercise preferences. What is the largest population to which the results of this survey can be generalized?
Show the answer and explanation
Why C is right
This is a convenience sample of only those members who attended a specific class at a specific time. The results can only be generalized to members who attend that particular 6:00 AM Monday spin class, not to all members or even all spin class attendees.
Why the others are wrong
AThis overgeneralizes by assuming the convenience sample represents all fitness center members, when many members may prefer different times, different classes, or different activities.
BThis overgeneralizes by ignoring that the sample was taken from only one specific time slot, so it cannot represent all spin class attendees across different times and days.
DThis assumes the 6:00 AM Monday spin class represents all early morning classes, but different classes at different times may attract members with different preferences.
Question 21Hard
A market research firm conducted two independent surveys about consumer preferences. Survey A sampled 625 randomly selected consumers and found that 56% prefer Brand X, with a margin of error of±4 percentage points at a 95% confidence level. Survey B sampled a different number of randomly selected consumers from the same population and found that 51% prefer Brand X, with a margin of error of±2 percentage points at the same confidence level. How many consumers were sampled in Survey B?
Show the answer and explanation
Why C is right
The margin of error in Survey B (±2 percentage points) is half that of Survey A (±4 percentage points). Since margin of error is inversely proportional to the square root of sample size, halving the margin requires quadrupling the sample size. Therefore, Survey B sampled 625×4=2500 consumers.
Why the others are wrong
AThis incorrectly assumes that halving the margin of error requires doubling the sample size, but the relationship involves the square root, so a factor of 4 is needed.
BWhile the confidence intervals do overlap, the question asks about the relationship between sample sizes and margins of error, which can be determined from the given information.
DThis incorrectly assumes that halving the margin of error means halving the sample size, reversing the actual relationship between these quantities.
Question 22Hard
Poll A surveyed 1200 randomly selected adults and found that 38% support a proposed regulation, with a margin of error of±3 percentage points. Poll B surveyed 300 randomly selected adults from the same population and found that 42% support the regulation. Assuming margin of error is inversely proportional to the square root of sample size, what can be concluded about the two polls?
Show the answer and explanation
Why A is right
Poll B has one-fourth the sample size of Poll A, so its margin of error is doubled: ±3%×2=±6 percentage points. Poll A's CI is 35% to 41%, and Poll B's CI is 36% to 48%. These intervals overlap from 36% to 41%, so the results are statistically indistinguishable.
Why the others are wrong
BThis ignores the margins of error and only compares point estimates, which is insufficient for determining whether results are statistically different.
CThis incorrectly halves the margin of error when it should double, reversing the effect of decreased sample size.
DPoll A's CI is correctly stated as 35% to 41%, and Poll B's as 36% to 48%, but these intervals do overlap, so the conclusion of statistical difference is incorrect.
Question 23Hard
Two independent surveys were conducted about support for a local tax measure. Survey X sampled 5000 randomly selected voters and found 61% in favor, with a margin of error of±1.5 percentage points at a 95% confidence level. Survey Y sampled 1250 randomly selected voters from the same population and found 58% in favor. Assuming the same confidence level and methodology, what is the approximate margin of error for Survey Y?
Show the answer and explanation
Why D is right
Survey Y has a sample size of 1250, which is 1/4 of Survey X's 5000. When sample size decreases by a factor of 4, the margin of error increases by a factor of 4=2. Therefore, Survey Y's margin of error is 1.5%×2=±3 percentage points.
Why the others are wrong
AThis presents confidence intervals rather than margins of error. While these calculations might correspond to the margins, the question specifically asks for the margin of error for Survey Y.
BThis incorrectly assumes that reducing the sample size by a factor of 4 reduces the margin of error by a factor of 2. In reality, smaller samples have larger margins of error.
CThis provides the correct numerical value but drops the ± notation, which is essential for expressing margin of error as it indicates the estimate could be that amount above or below the point estimate.
Question 24Hard
A survey of randomly selected smartphone users found that 58% use their phone for more than 3 hours per day, with a margin of error of±4 percentage points. Another survey from the same population with four times as many respondents found that 62% use their phone for more than 3 hours per day. Based on these results, which of the following is true?
Show the answer and explanation
Why C is right
The first survey has a confidence interval of 54% to 62%. The second survey, with four times the sample size, has a margin of error of ±2 percentage points (±4÷2), giving an interval of 60% to 64%. These intervals overlap from 60% to 62%, indicating the difference may not be statistically significant.
Why the others are wrong
AThis compares only the point estimates without considering the margin of error, which is essential for determining statistical significance.
BThis incorrectly suggests the margin of error increases when sample size increases; it actually decreases to ±2 percentage points.
DThis misinterprets what margin of error indicates; a smaller margin of error means more precision, but both surveys can be valid for comparison using their respective confidence intervals.
Question 25Hard
A transportation study surveyed 3200 randomly selected commuters and found that 45% use public transit, with a margin of error of±1.5 percentage points for a 95% confidence level. Researchers want to conduct a follow-up survey with the same methodology but can only survey 800 randomly selected commuters. What will be the approximate margin of error for the follow-up survey at the same confidence level?
Show the answer and explanation
Why D is right
The follow-up survey will have a sample size of 800, which is 1/4 of the original 3200. When sample size decreases by a factor of 4, the margin of error increases by a factor of 4=2. Therefore, the margin of error will be 1.5%×2=±3 percentage points.
Why the others are wrong
AThis drops the ± notation and incorrectly suggests the margin of error remains unchanged despite the sample size decreasing by a factor of 4.
BThis incorrectly assumes that reducing the sample size by a factor of 4 reduces the margin of error by a factor of 2. The relationship is inverse: smaller samples produce larger margins of error.
CThis presents a confidence interval rather than the margin of error. Additionally, the interval shown uses approximately ±3.5 percentage points, not the correct ±3.
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