Inference & margin of error
When researchers measure a sample, they use it to estimate something about the whole population. The margin of error tells you how far off that estimate might be — so the population value lies in a range, not at a single number.
What College Board tests
Building the interval of plausible values from an estimate and its margin of error, estimating a population total from a sample percentage, interpreting what a margin of error does and does not tell you, and knowing what makes a margin of error larger or smaller.
The core ideas
The margin of error describes the population value, not the sample. The sample's measured percentage is known exactly; it's the population value that lives in the plus-or-minus range.
Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.
1 · Build the plausible-value interval
A random sample of a town's residents was surveyed. An estimated 42% support a proposed park, with a margin of error of 3%. Which of the following is most likely to be the actual percent of all residents who support the park?
Approach Plausible values lie within the margin of error of the estimate. Build the interval from to , then see which choice falls inside it.
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Answer: B
Only 44% falls inside the interval from 39% to 45%, so it's the only plausible value among the choices.
Why the other choices are wrong
A 38% is just below the lower bound of 39%.
C 47% is above the upper bound of 45%.
D 52% is far outside the interval.
2 · Estimate a population total
In a random sample, 38% of respondents reported owning a bicycle. If the town has 2,500 residents, which is the best estimate of the number of residents who own a bicycle?
Approach A sample percentage estimates the same percentage of the whole population. Apply 38% to the full 2,500 residents — "percent of" means multiply.
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Answer: B
Why the other choices are wrong
A Applies only 15.2% (or mis-scales by a factor of 10).
C Estimates the non-owners (62%) instead.
D Uses the whole population as if everyone owns a bicycle.
3 · Interpret the margin of error
A study estimates that 30% of a city's teens read daily, with a margin of error of 3%. Which statement is best supported?
- Exactly 30% of the city's teens read daily.
- It is plausible that the true percent of the city's teens who read daily is between 27% and 33%.
- The percent of teens who read daily is definitely 27%.
- 3% of the city's teens do not read daily.
Approach The margin of error gives a plausible range for the population value, not an exact figure and not a statement about a specific 3%. Build the interval and read the choices against what a margin of error actually claims.
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Answer: B
A margin of error never pins down an exact value or guarantees one — it gives a plausible range for the population percentage. Choice B states exactly that.
Why the other choices are wrong
A "Exactly" overclaims — the 30% is an estimate, not certain.
C "Definitely" overclaims, and 27% is just one end of the range.
D Misreads the 3% as a count of teens rather than a margin of error.
4 · What changes the margin of error
Two surveys estimate the same quantity using random samples selected from the same population, with margins of error calculated using the same method. Survey X samples 400 people; Survey Y samples 1,600 people. Which survey would be expected to have the smaller margin of error, and why?
- Survey X, because a smaller sample is easier to measure.
- Survey Y, because a larger random sample gives a more precise estimate.
- They would be equal, because the population is the same.
- It cannot be determined from the information given.
Approach For random samples, the margin of error depends on sample size: more data narrows the range. Compare the two sample sizes and pick the larger one.
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Answer: B
A larger random sample reduces the effect of chance variation, so its estimate is more precise and its margin of error is smaller. Survey Y's 1,600 beats Survey X's 400.
Why the other choices are wrong
A Smaller samples are less precise, not easier in the relevant sense.
C Equal populations don't make equal margins of error — sample size differs.
D The sample sizes are given, so it can be determined.