SAT Inference from samples & margin of error
Interpret estimates, margins of error, and what a sample can claim.
How to score it
- Estimate ± margin of error gives a plausible interval — read it as “between (est − m) and (est + m).”
- Scale a sample proportion to the population by multiplying by population size.
- A larger sample shrinks the margin of error.
Common traps
- 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.
The 4 question types, with real examples
Interpret estimate ± margin of error
“Which is the best interpretation of the survey results?”
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?
- AIt is plausible that between 60.8% and 67.2% of all adults exercise at least three times per week.✓
- BThe margin of error indicates that 3.2% of the 900 adults surveyed were uncertain about their exercise habits.
- CIt is plausible that more than 67.2% of all adults exercise at least three times per week.
- DIf the sample size were reduced to 450 adults, the margin of error would decrease to approximately 1.6%.
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.
Population estimate from a sample
“What is the best estimate of the number in the population who … ?”
From a population of 80000 residents, 800 were chosen at random and surveyed about whether they use public transportation regularly. Based on the survey, it is estimated that 28% of residents in the population use public transportation regularly, with an associated margin of error of 3%. Based on these results, which of the following is a plausible value for the total number of residents in the population who use public transportation regularly?
- A
- B✓
- C
- D
The confidence interval ranges from 28% - 3% = 25% to 28% + 3% = 31%. Applying 26% (a value within this range) to the population: 0.26 × 80000 = 20800, which is plausible.
Best generalization population
“[sampling method described]. What is the largest population to which the results of this survey can be generalized?”
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?
- AAll students at Lincoln High School
- BAll senior students at Lincoln High School who are enrolled in AP Biology✓
- CAll students enrolled in AP Biology courses nationwide
- DAll senior students at Lincoln High School
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.
Compare margin of error across two samples
“[survey 1 with sample size n1, margin of error m1]. [survey 2 / competing poll]. Which statement correctly compares the two, or explains why one has a smaller margin of error?”
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?
- AThe margin of error would be reduced to approximately 1.25%.
- BThe margin of error would remain approximately 2.5%.
- CThe margin of error would increase to approximately 5%.✓
- DThe margin of error would increase to approximately 10%.
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 , so it would double to approximately 5%.
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