20 medium SAT Inference from samples & margin of error questions
Medium is where most scores are actually won and lost. These questions are not tricky for the sake of it, but every one of them has a wrong answer built to catch a specific shortcut.
Every question below is a real item from the SAT Climb bank, tagged medium by the same difficulty model the app uses to build your practice. Pick an answer before you open the explanation.
Math · Problem-Solving and Data Analysis~1 per testMedium tier
A survey of 800 randomly selected registered voters found that 58% support a proposed policy change, with an associated margin of error of 3.5%. Based on this survey, which of the following is the most appropriate conclusion about the proportion of all registered voters who support the proposed policy change?
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Why A is right
The margin of error of 3.5% means the true proportion is plausibly within 3.5 percentage points above or below the sample estimate. Subtracting 3.5% from 58% gives 54.5%, and adding 3.5% gives 61.5%, so the interval of plausible values is 54.5% to 61.5%.
Why the others are wrong
BThis choice incorrectly uses the margin of error as an endpoint of the interval rather than the amount to add and subtract from the estimate.
CThis choice only uses the lower bound and suggests values below it are plausible, failing to recognize that the interval includes values above the estimate as well.
DThis choice incorrectly suggests the margin of error increases with sample size, when in fact larger samples produce smaller margins of error.
Question 2Medium
From a population of 120000 households, 1200 were chosen at random and surveyed about their recycling practices. Based on the survey, it is estimated that 42% of households in the population regularly recycle, with an associated margin of error of 2.8%. Based on these results, which of the following is a plausible value for the total number of households in the population that regularly recycle?
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Why C is right
The margin of error of 2.8% means the true proportion is plausibly between 39.2% and 44.8%. Calculating 40% of 120000 gives 48000, which falls within the plausible interval, making it a plausible value for the total.
Why the others are wrong
AThis choice incorrectly calculates 2.8% of 120000, confusing the margin of error itself with the interval of plausible proportions.
BThis choice uses only the point estimate of 42% without accounting for the margin of error, which would give 50400 households, not considering the range of plausible values.
DThis choice incorrectly suggests that increasing sample size would change the estimated proportion itself, when sample size affects margin of error, not the point estimate.
Question 3Medium
A researcher surveyed all 240 students in the senior class at Lincoln High School who are enrolled in AP Biology about their study habits. What is the largest population to which the results of this survey can be generalized?
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Why B is right
The survey sampled all students meeting two criteria: being seniors at Lincoln High School AND being enrolled in AP Biology. The results can only be generalized to the exact population from which the complete census was taken, which is senior students at Lincoln High School who are enrolled in AP Biology.
Why the others are wrong
AThis overgeneralizes by ignoring the AP Biology enrollment criterion; the survey only included seniors in AP Biology, not all students.
CThis overgeneralizes beyond the school; results from one school cannot be extended to all schools nationwide without random sampling from that broader population.
DThis overgeneralizes by ignoring the AP Biology enrollment requirement; only seniors enrolled in AP Biology were surveyed.
Question 4Medium
A polling organization surveyed a random sample of 800 likely voters. Based on the sample, it is estimated that 44% support Candidate A, with an associated margin of error of 3.5%. A separate random sample of 800 likely voters estimated that 48% support Candidate B, with the same margin of error of 3.5%. Based on these results, which of the following is the most appropriate conclusion?
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Why B is right
Candidate A's confidence interval is 40.5% to 47.5%, and Candidate B's is 44.5% to 51.5%. These intervals overlap in the region from 44.5% to 47.5%, so it is plausible that the true support levels are equal within this overlap region.
Why the others are wrong
AThis choice ignores the margins of error and treats the sample estimates as exact population values, failing to account for sampling variability.
CThis choice incorrectly applies the margin of error in only one direction for Candidate A, creating an incomplete confidence interval.
DThis choice reverses the relationship between sample size and margin of error; reducing sample size would increase margins of error, leading to more overlap, not less.
Question 5Medium
A sample consisting of 900 college students was selected at random for a study. Based on the sample, it is estimated that 74% of all college students have part-time jobs, with an associated margin of error of 2.9%. Which of the following is the most plausible conclusion about all college students?
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Why D is right
Subtracting the margin of error from the estimate gives 74% - 2.9% = 71.1%, and adding the margin of error to the estimate gives 74% + 2.9% = 76.9%. Therefore, it is plausible that between 71.1% and 76.9% of all college students have part-time jobs.
Why the others are wrong
AThis choice misunderstands the purpose of random sampling and margin of error, which allow conclusions to be drawn about populations from samples.
BThis choice uses only the lower bound and suggests all values are below it, rather than recognizing the full interval as plausible.
CThis choice incorrectly suggests that increasing sample size increases margin of error, when in fact larger samples produce smaller margins of error.
Question 6Medium
A health researcher surveyed 900 randomly selected adults and found that 64% exercise at least three times per week, with an associated margin of error of 3.2%. Based on this survey, which of the following statements is most appropriate?
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Why A is right
The estimate of 64% with a margin of error of 3.2% creates a confidence interval from 60.8% to 67.2%. This interval represents the range of plausible values for the true proportion of all adults who exercise at least three times per week.
Why the others are wrong
BThis choice misinterprets the margin of error as relating to survey respondent uncertainty rather than statistical precision of the population estimate.
CThis choice incorrectly suggests values above the upper bound of the confidence interval are plausible, when the margin of error defines the upper limit at 67.2%.
DThis choice incorrectly claims that reducing sample size would decrease the margin of error, when smaller samples actually produce larger margins of error.
Question 7Medium
From a population of 80000 employees at a large company, 1200 were chosen at random and surveyed about remote work preferences. Based on the survey, it is estimated that 42% of employees in the population prefer remote work, with an associated margin of error of 2.8%. Based on these results, which of the following is a plausible value for the total number of employees in the population who prefer remote work?
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Why C is right
The confidence interval for the proportion is 0.42±0.028, giving a range of 0.392 to 0.448. Multiplying by the population of 80000 gives a range of 31360 to 35840 employees. The value 32400 falls within this plausible range.
Why the others are wrong
AThis uses only the sample size (1200 × 0.42) instead of the full population.
BThis incorrectly uses only the margin of error (0.028 × 80000) as the answer.
DThis uses only the upper bound of the confidence interval (0.448 × 80000 = 35840) and then incorrectly inflates it further.
Question 8Medium
A transportation agency randomly selected 320 commuters from a list of all individuals who use public transportation at least three times per week in Metro Area Y and surveyed them about transit improvements. What is the largest population to which the results of this survey can be generalized?
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Why D is right
The sampling frame included individuals meeting two criteria: using public transportation at least three times per week and residing in Metro Area Y. Random sampling from this frame allows generalization to all individuals who use public transportation at least three times per week in Metro Area Y.
Why the others are wrong
AThis includes individuals who use public transportation less frequently than three times per week, but the sampling frame required at least three times per week.
BThis includes all commuters regardless of whether they use public transportation, but the sampling frame was limited to public transportation users.
CThis overgeneralizes beyond Metro Area Y to a nationwide population that was not part of the sampling frame.
Question 9Medium
A random sample of 900 college students was selected to estimate the proportion who work part-time jobs. Based on the sample, it is estimated that 64% of all college students work part-time jobs, with an associated margin of error of 3.2%. If the sample size were increased to 3,600 students, which of the following best describes how the margin of error would change?
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Why A is right
The margin of error is inversely proportional to the square root of the sample size. Increasing the sample size by a factor of 4 (from 900 to 3,600) decreases the margin of error by a factor of 4=2, so it would be halved to approximately 1.6%.
Why the others are wrong
BThis incorrectly suggests that increasing sample size increases the margin of error, which is the opposite of the true relationship.
CThis incorrectly suggests that changing the sample size has no effect on the margin of error.
DThis applies the factor of 4 directly to the margin of error instead of using the square root relationship.
Question 10Medium
A random sample of 1200 adult smartphone users was surveyed about app usage. Based on the sample, it is estimated that 51% of all adult smartphone users have more than 30 apps installed, with an associated margin of error of 2.8%. Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of all adult smartphone users who have more than 30 apps installed?
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Why D is right
Subtracting the margin of error from the estimate gives 0.51 - 0.028 = 0.482, and adding the margin of error to the estimate gives 0.51 + 0.028 = 0.538. Therefore, it is plausible that the proportion is between 0.482 and 0.538.
Why the others are wrong
AThis choice considers only values above the upper bound of the confidence interval, ignoring that the entire interval contains plausible values.
BThis choice states the interval correctly but uses language that confuses the margin of error with the confidence interval itself, treating it as definitive rather than indicating plausibility.
CThis choice reverses the relationship between sample size and margin of error; decreasing sample size increases margin of error.
Question 11Medium
A random sample of 1200 households was selected to estimate the proportion of households in a county that have solar panels. Based on the sample, it is estimated that 18% of households have solar panels, with an associated margin of error of 2.2%. Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of households in the county that have solar panels?
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Why A is right
The confidence interval is calculated by subtracting and adding the margin of error: 0.18 - 0.022 = 0.158 and 0.18 + 0.022 = 0.202. Therefore, it is plausible that the proportion is between 0.158 and 0.202.
Why the others are wrong
BThis choice treats the sample estimate as the exact population parameter, ignoring the sampling variability indicated by the margin of error.
CThis choice suggests values below the lower bound of the confidence interval are plausible, which contradicts the definition of the confidence interval.
DThis choice incorrectly suggests the margin of error scales linearly with sample size, when in fact it decreases as sample size increases (roughly proportional to the inverse square root of sample size).
Question 12Medium
A medical researcher obtained a list of all patients aged 50-65 who visited Riverside Clinic in 2023 for annual checkups and randomly selected 200 of them for a health survey. What is the largest population to which the results of this survey can be generalized?
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Why D is right
The sampling frame consisted of patients meeting three criteria: aged 50-65, visited Riverside Clinic in 2023, and came for annual checkups. Since random sampling was used from this frame, results can be generalized to all patients aged 50-65 who visited Riverside Clinic in 2023 for annual checkups.
Why the others are wrong
AThis ignores both the age restriction (50-65) and the visit purpose (annual checkups); the sampling frame had these specific qualifiers.
BThis overgeneralizes beyond Riverside Clinic to a nationwide population that was not part of the sampling frame.
CThis extends beyond Riverside Clinic to other clinics, but the sampling frame was limited to one clinic.
Question 13Medium
A survey of 1200 adults was conducted to estimate the proportion of adults in a state who have visited a national park in the past year. Based on the survey, it is estimated that 42% of all adults in the state have visited a national park in the past year, with an associated margin of error of 2.8%. If the sample size were doubled to 2400 adults, which of the following best describes how the margin of error would change?
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Why C is right
The margin of error is inversely proportional to the square root of the sample size. Doubling the sample size multiplies the sample size by 2, so the margin of error is divided by the square root of 2, which is approximately 1.41. Therefore, 2.8% ÷ 1.41 ≈ 2.0%.
Why the others are wrong
AThis incorrectly assumes the margin of error is inversely proportional to the sample size rather than to the square root of the sample size.
BThis incorrectly suggests that the margin of error does not change when sample size changes.
DThis incorrectly multiplies the margin of error by 2 instead of dividing it by the square root of 2.
Question 14Medium
A random sample of 600 small business owners was surveyed about their hiring plans. Based on the sample, it is estimated that 27% of all small business owners plan to hire additional employees in the next quarter, with an associated margin of error of 3.7%. Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of all small business owners who plan to hire additional employees?
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Why C is right
Subtracting the margin of error from the estimate gives 0.27 - 0.037 = 0.233, and adding the margin of error to the estimate gives 0.27 + 0.037 = 0.307. Therefore, it is plausible that the proportion is between 0.233 and 0.307.
Why the others are wrong
AThis choice only considers values above the upper bound, ignoring that the entire interval from 0.233 to 0.307 contains plausible values.
BThis choice treats the estimate as exact rather than understanding that the margin of error creates an interval of plausible values.
DThis choice incorrectly suggests that decreasing sample size would decrease margin of error, when in fact it would increase.
Question 15Medium
A random sample of 500 registered voters was selected from a district. Based on the sample, it is estimated that 58% of all registered voters in the district support a proposed policy, with an associated margin of error of 4%. Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of all registered voters in the district who support the proposed policy?
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Why A is right
The confidence interval is formed by subtracting and adding the margin of error to the estimate: 58% - 4% = 54% and 58% + 4% = 62%. Therefore, it is plausible that the proportion is between 0.54 and 0.62.
Why the others are wrong
BThis choice incorrectly applies the margin of error in only one direction (downward), dropping the upper bound that results from adding the margin of error.
CThis choice correctly identifies the interval but incorrectly states that a smaller sample size would produce a narrower interval. In fact, smaller samples produce wider margins of error.
DThis choice confuses the margin of error with a direct measure of voter uncertainty or indecision, rather than understanding it as a measure of sampling variability.
Question 16Medium
A university researcher randomly selected 180 undergraduate students from the complete enrollment list of students majoring in engineering who are in their junior year at State University. What is the largest population to which the results of this survey can be generalized?
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Why D is right
The sampling frame included students meeting all three criteria: undergraduate status, engineering major, and junior year at State University. Random selection from this frame allows generalization to all undergraduate students majoring in engineering who are in their junior year at State University.
Why the others are wrong
AThis includes engineering majors from all years (freshmen, sophomores, seniors), but the sampling frame was restricted to juniors only.
BThis includes all majors, but the sampling frame was limited to engineering majors only.
CThis overgeneralizes to universities nationwide, but sampling was only from State University.
Question 17Medium
A random sample of 1,600 voters was surveyed about a proposed tax increase. Based on the sample, it is estimated that 39% of all voters oppose the increase, with an associated margin of error of 2.5%. If the sample size were reduced to 400 voters, which of the following best describes how the margin of error would change?
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Why C is right
The margin of error is inversely proportional to the square root of the sample size. Reducing the sample size by a factor of 4 (from 1,600 to 400) increases the margin of error by a factor of 4=2, so it would double to approximately 5%.
Why the others are wrong
AThis incorrectly suggests that decreasing sample size decreases the margin of error, which is the opposite of the true relationship.
BThis incorrectly suggests that changing the sample size has no effect on the margin of error.
DThis applies the factor of 4 directly to the margin of error instead of using the square root relationship.
Question 18Medium
A random sample of 800 registered voters was surveyed about a ballot measure. Based on the sample, it is estimated that 58% of all registered voters support the measure, with an associated margin of error of 3.5%. Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of all registered voters who support the measure?
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Why A is right
Subtracting the margin of error from the estimate gives 0.58 - 0.035 = 0.545, and adding the margin of error to the estimate gives 0.58 + 0.035 = 0.615. Therefore, it is plausible that the proportion is between 0.545 and 0.615.
Why the others are wrong
BThis choice confuses the margin of error with the confidence interval by rounding the bounds, suggesting uncertainty about what the interval represents.
CThis choice only applies the margin of error in one direction, dropping the upper bound of the interval.
DThis choice reverses the relationship between sample size and margin of error; increasing sample size decreases margin of error.
Question 19Medium
A researcher surveyed a random sample of 600 small business owners. Based on the sample, it is estimated that 67% of all small business owners plan to expand their operations within the next year, with an associated margin of error of 3.8%. Which of the following is the most plausible conclusion about all small business owners?
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Why C is right
The confidence interval is calculated by subtracting and adding the margin of error to the estimate: 67% - 3.8% = 63.2% and 67% + 3.8% = 70.8%. Therefore, it is plausible that between 63.2% and 70.8% of all small business owners plan to expand.
Why the others are wrong
AThis choice only applies the margin of error in the positive direction, failing to include the lower bound of the confidence interval.
BThis choice treats the sample estimate as an exact population parameter, ignoring the uncertainty represented by the margin of error.
DThis choice incorrectly suggests that increasing sample size would increase the margin of error, when in fact larger samples produce smaller margins of error.
Question 20Medium
A city planning department randomly selected 150 registered voters from a list of all registered voters who live in the downtown district and asked them about their opinions on a new park proposal. What is the largest population to which the results of this survey can be generalized?
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Why C is right
The sampling frame was all registered voters who live in the downtown district, and a random sample was drawn from this frame. The results can be generalized to the entire sampling frame: all registered voters who live in the downtown district.
Why the others are wrong
AThis includes non-registered voters in the downtown district, but the sampling frame only included registered voters, so results cannot be generalized to those not on the voter registration list.
BThis ignores the geographic restriction; the sampling frame was limited to the downtown district, not the entire city.
DThis overgeneralizes beyond the single city; random sampling was only done within one city's downtown district.
What to do after medium
Medium is the tier that decides most scores. If these are landing, the hard set is where the remaining points are.
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