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Inference from sample statistics and margin of error: SAT Math notes
These questions test whether you can use a random sample to estimate something about a whole population, and interpret what a margin of error does and doesn't tell you.
Study it with worked examples and 8 practice questionsPractise Inference from sample statistics and margin of error questions
What this topic tests
It's usually impossible to ask everyone, so researchers ask a random sample and use the result to estimate the whole population. If 20 of 80 randomly chosen customers bought fruit, the best estimate is that about 25% of all the customers did.
Because a sample isn't the whole population, the estimate comes with a margin of error. The estimate plus or minus the margin of error gives a range of plausible values for the true population value. Values inside the range are plausible; values outside it are doubtful, but not impossible.
The SAT tests the meaning of these ideas more than the arithmetic. Know that a margin of error describes the population's mean or percent, not each individual, that larger random samples give smaller margins of error, and that a conclusion only covers the population the sample was drawn from.
Key ideas
- Population estimate = sample proportion × population size, or sample mean × number of units.
- Plausible range = from (estimate minus margin of error) to (estimate plus margin of error).
- The margin of error is the distance from the estimate to either end of the range, which is half the range's width.
- The range is about the population's mean or percent. It does not say that every individual falls inside it.
- Plausible means likely, not certain. A value outside the range is doubtful, not impossible.
- With the same method, a larger random sample gives a smaller margin of error, and a smaller sample gives a larger one.
- A sample only tells you about the population it was randomly drawn from, such as the largemouth bass in one pond, not all fish.
The 3 kinds of Inference from sample statistics and margin of error question
Sorted from every Inference from sample statistics and margin of error question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Scale a sample up to the population
How to spot it. A random sample gives a count or a mean, and you're asked for the best estimate for a whole group: how many council members support the bill, how many trees are in the forest, how many voters plan to vote for Candidate A.
How to answer it:
- Find the sample proportion (count ÷ sample size) or the sample mean.
- Multiply by the size of the whole population: the number of people, acres, classes, and so on.
- If the question asks about the opposite group (fewer than two siblings, say), subtract the proportion from 1 first.
- Remember it's an estimate. Choices that say exactly or must are too strong.
The trap. Stopping at the sample's count, or treating the estimate as exact. If 20 of 80 sampled customers bought fruit, the answer isn't 20: scale it to all 1,500 customers. And a choice that says exactly 375 customers must have bought fruit is wrong even though 375 is the right number.
2. Use and interpret a margin of error
How to spot it. You're given an estimate and a margin of error (or a range) and must build the plausible range, find the margin of error from the range, turn the range into counts, or pick the statement that interprets it correctly.
How to answer it:
- Build the range: estimate minus the margin to estimate plus the margin.
- If you're given the range instead, the margin of error is the distance from its center (the estimate) to either end.
- If the question wants counts, multiply both ends of a percent range by the population size.
- Check the wording of the choices: the range is about the population's mean or percent, it gives plausible values (not certain ones), and it says nothing about individuals.
The trap. Choices that apply the range to every individual (all visitors hiked between 4 and 5 miles) or that turn plausible into certain (it is not possible that fewer than 27% are heavy texters). A margin of error only tells you where the population's mean or percent probably is.
3. Sample size and who the results describe
How to spot it. The question asks why one margin of error is bigger than another, what happens to the margin of error with a larger sample, or which conclusion the data support when the sample came from only part of a population.
How to answer it:
- Decide which kind of question it is: one about the size of the margin of error, or one about who the results describe.
- Size: with the same method, a larger random sample gives a smaller margin of error, and a smaller sample gives a larger one. Nothing else in the choices matters.
- Who the results describe: find exactly which group the sample was drawn from (for example, largemouth bass in one pond) and choose the conclusion about that group only.
The trap. Thinking a bigger sample makes the margin of error bigger, or pushes the estimate itself up or down. A bigger sample just makes the estimate more precise. In scope questions, avoid choices that stretch a sample of largemouth bass into a claim about all the fish in the pond.
Practise Inference from sample statistics and margin of error
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