SAT Math — margins of error, sample size and overlapping estimates

SAT Math includes a skill called inference from sample statistics and margin of error. College Board's description asks students to use a sample mean or sample proportion to estimate a population mean or proportion, to interpret a margin of error, and to understand that a larger sample generally leads to a smaller margin of error. On the test these questions are short and wordy rather than computational, and the hard part is knowing exactly what a margin of error does and does not allow you to say. This material concentrates on that.

Every percent here comes with a margin stated in percentage points, which is how polls report it, and one question checks the difference directly: 40 percent plus or minus 3 points is 37 to 43 percent, not 3 percent of 40. From there you build ranges of plausible values and use them. You pick the one plausible population mean for nightly sleep, find the value that falls outside a store's range, turn a range of percents into a range of residents in a town of 90,000, and turn a range for a mean purchase into a range for the total spent by 6,000 customers.

Several questions compare two estimates. A four-point lead with a margin of 3 points on each side leaves the two ranges overlapping, so the poll cannot name a leader; a gap between 58 and 36 percent with a margin of 4 points leaves the ranges apart, so it can. The same reasoning applies over time: a rise from 41 to 44 percent between two polls with margins of 3.5 points shows no real change.

The remaining questions are about sample size and sampling method. You predict that quadrupling a sample from 400 to 1,600 gives a smaller margin, decide which of two samples is more precise, and choose the change to a study that would narrow its margin. An online poll of 5,000 self-selected readers with a tiny reported margin closes the set: a margin of error measures only the chance variation of random samples, and says nothing about bias.

The explanations show every range and name the misreading behind each wrong answer, such as treating a sample value as exact or applying a percent range to the sample instead of the population. The flashcards summarize margins of error in percentage points, the effect of sample size, overlapping and separate ranges, plausible counts and totals, and what a margin cannot measure.

The material offers a quiz and flashcards. It is independent practice; all polls and studies are invented, and no question is taken from any published test.

• Build the range of plausible population values from an estimate and a margin of error in percentage points
• Distinguish a margin in percentage points from a percent of the estimate
• Decide whether two estimates, or two polls over time, show a real difference by checking whether their ranges overlap
• Explain that a larger random sample generally gives a smaller margin of error
• Convert a range for a percent or a mean into a range for a count or a total
• Explain why a margin of error does not account for bias from a nonrandom sample

Practice material written by Zestly, based on the SAT Math skill "Inference from sample statistics and margin of error" in the Problem-Solving and Data Analysis domain, as described in College Board's Assessment Framework for the Digital SAT Suite (version 3.01, August 2024, Appendix B). All data sets and situations are invented. Zestly is not affiliated with College Board.