PSAT Data — from a random sample to the population

Inference from sample statistics is one of the skills College Board lists for Problem-Solving and Data Analysis on the PSAT/NMSQT. On the test it usually looks like this: a random sample of some size shows a certain proportion or mean, and you are asked what that says about the whole population, or about which population it says anything at all. This material practices exactly that, and stops there: margins of error and the distinction between observational studies and experiments are treated in College Board's description as SAT content, so they are not included.

The twelve quiz questions split into two groups. Six are estimates. A random sample of households gives a proportion to apply to the whole town, where the wrong options come from applying it to the sample, using the wrong sample size, or reading a count as a percent. A shipment of light bulbs gets an expected number of defectives. A coffee survey asks for the number who do NOT prefer a brand, so the complement matters. A sample mean of people per household becomes an estimate of a town's population, a percent of online buyers becomes a count, and a capture-recapture study estimates the fish in a lake from the share of tagged fish in a later catch.

The other six are about what a sample can and cannot tell you. A random sample from one county's voter list describes that county's registered voters, not the state's. A sample drawn from a store's database of last year's buyers describes those buyers. One question asks for the best way to choose a sample of employees, one explains why an estimate from a random sample is close but not exact, and two show sampling going wrong: a gym survey handed out only at 6 a.m., and a random sample drawn from a list of landline owners, which leaves out everyone who does not have one.

The flashcards cover the vocabulary: population, sample, random sample, sample proportion, estimating counts and totals, the population a result applies to, sampling bias, convenience and voluntary samples, and capture-recapture.

The printable written work has eight open problems: bikers and non-bikers in a company, a deer population from tagging, a complex's daily water use, the scope of a sample of nurses, three sampling plans for a library compared, two honest samples that disagree, a count estimate to report sensibly, and a huge online poll set against a small random sample.

The oral exam asks one question at a time and wants the reasoning: which proportion, which population, and why.

All situations are invented for practice, and nothing is taken from any published test.

• Estimate a population count from the proportion in a random sample
• Estimate a population total from a sample mean
• Use the complement of a sample proportion
• Estimate a population size by capture-recapture
• Identify the population to which a random sample's results apply
• Explain why convenience samples, voluntary responses and incomplete lists can bias results
• Explain why an estimate from a random sample is approximate rather than exact

Practice material written by Zestly, based on the PSAT/NMSQT Math skill "Inference from sample statistics" as described in College Board's Fall 2026 PSAT/NMSQT Student Guide and the Assessment Framework for the Digital SAT Suite (version 3.01, August 2024). All situations are invented. Zestly is not affiliated with College Board.