Inference & Margin of Error
When you survey 800 people and ask whether they support a candidate, you don't actually learn what the country thinks — you learn what those 800 people think. Inference is the process of using the sample to make a careful, qualified guess about the broader population. The SAT tests inference through one specific tool: the margin of error. The hardest questions don't ask you to compute anything — they ask which interpretations of a confidence interval are correct, and the wrong interpretations sound very convincing.
Samples, populations, and the gap between them
Two terms you need to keep distinct:
- Population: the entire group you want to know about (e.g., all 50 million eligible voters in a country).
- Sample: the subset of the population that's actually surveyed or measured (e.g., 800 randomly selected voters).
Statistics computed from the sample are estimates of the corresponding population values:
| Sample statistic | Estimates the population parameter |
|---|---|
| Sample mean (x̄) | Population mean (μ) |
| Sample proportion (p̂) | Population proportion (p) |
Sample statistics are almost never EXACTLY equal to the population parameter — they're approximations. The margin of error tells you how close that approximation likely is.
Margin of error: what it actually says
A margin of error gives you a range of plausible values for the population parameter, based on the sample statistic. It's typically reported alongside a confidence level (most often 95%).
If a survey reports "47% of voters support Candidate A, with a margin of error of ±3 percentage points at the 95% confidence level," the confidence interval is:
What "95% confident" actually means
This is where the SAT writes its hardest questions and the most tempting wrong answers. Most students have a casual understanding of "95% confident" that turns out to be subtly wrong.
"47% ± 3% with 95% confidence" does NOT mean:
- 95% of voters fall in the range 44% to 50%
- 95% of all polls would give exactly this answer
- There's a 95% probability the true value is exactly 47%
What it DOES mean:
- If the polling were repeated many times with the same method, about 95% of the resulting confidence intervals would contain the true population proportion.
- For SAT purposes, the safe interpretation: we are 95% confident that the true population proportion lies between 44% and 50%.
Right and wrong interpretations
Given a survey result like "47% with a margin of error of ±3% at the 95% confidence level," here are the most common interpretations and whether they're valid:
- "We are 95% confident that the proportion of all voters who support Candidate A is between 44% and 50%."
- "The plausible range for the true population proportion is 44% to 50%."
- "The best single estimate of the population proportion is 47%."
- "95% of all voters support Candidate A at a level between 44% and 50%." (Makes a claim about individuals, not the proportion.)
- "There is a 95% probability that exactly 47% of voters support Candidate A." (Probability claim about the sample, not the population.)
- "95% of the time, exactly 47% of voters will support Candidate A." (Claim about future behavior or other samples.)
What changes the margin of error
Two factors affect the size of the margin of error. The SAT tests both:
| Change | Effect on margin of error |
|---|---|
| Increase sample size | Smaller margin of error (more precise estimate) |
| Decrease sample size | Larger margin of error (less precise) |
| Increase confidence level (e.g., 95% → 99%) | Larger margin of error (wider interval) |
| Decrease confidence level (e.g., 95% → 90%) | Smaller margin of error (narrower interval) |
Bigger sample = more information = better estimate = smaller margin. If you survey 4,000 people instead of 400, you can be much more precise about the population.
Higher confidence = wider net. To be MORE sure your interval contains the true value, you have to make the interval LARGER. A 99% confidence interval is wider than a 95% confidence interval.
The SAT often gives a question like: "If the same study were redone with twice as many participants, the margin of error would..." → answer: decrease. Larger sample → smaller margin.
Sample SAT-style problems
A poll of 1,200 randomly selected adults reports that 62% favor a new policy, with a margin of error of ±2.5 percentage points at the 95% confidence level. Which interpretation is most appropriate?
- (A) Exactly 62% of all adults favor the new policy.
- (B) Between 59.5% and 64.5% of the 1,200 adults surveyed favor the policy.
- (C) We are 95% confident that between 59.5% and 64.5% of all adults favor the new policy.
- (D) The probability that any individual adult favors the policy is 62%.
- Compute the interval: 62% ± 2.5% = 59.5% to 64.5%.
- The interval describes the population, not the sample (eliminate B). It doesn't claim exactness (eliminate A). It's not about individual probabilities (eliminate D).
A researcher repeats a survey with the same methodology but increases the sample size from 500 participants to 2,000 participants. Compared to the original survey, the new survey will likely have:
- (A) A larger margin of error
- (B) A smaller margin of error
- (C) The same margin of error
- (D) A higher confidence level
- Larger sample size = more information about the population = smaller margin of error. Sample size doesn't change confidence level.
A survey of 800 randomly selected high school students estimates that 40% participate in a school sport, with a margin of error of ±3% at the 95% confidence level. Which conclusion is NOT supported by these results?
- (A) The best estimate of the proportion of all high school students who play a sport is 40%.
- (B) We are 95% confident that the proportion of all high school students who play a sport is between 37% and 43%.
- (C) Exactly 40% of all high school students play a sport.
- (D) The plausible range for the proportion of all high school students who play a sport is 37% to 43%.
- (A), (B), and (D) all describe the interval correctly.
- (C) claims an exact value — but the whole point of margin of error is that we DON'T know the exact value. The data only supports a range.
A poll uses a 95% confidence level and reports a margin of error of ±4 percentage points. If the same data were used to construct a 99% confidence interval instead, the margin of error would:
- (A) Stay the same
- (B) Decrease
- (C) Increase
- (D) Become zero
- Higher confidence level = wider interval (you need a bigger range to be MORE sure it contains the true value).
- So margin of error would increase.
1. Claiming exactness. "Exactly 47% of all voters support..." — never valid. The sample gives a single estimate, but only a range can be claimed about the population.
2. Confusing sample with population. The interval describes plausible values for the POPULATION proportion. Saying "47% ± 3% means between 44% and 50% of those surveyed support..." is wrong — the people surveyed gave the sample proportion of 47% exactly.
3. Inverting the sample size effect. Larger sample → SMALLER margin of error. Smaller sample → LARGER margin of error. Students sometimes think "more people sampled = more uncertainty," but it's the opposite — more data means more precision.