SAT Math — probability, and what a sample is allowed to say

Two ideas carry almost all the weight in this part of the test, and neither of them is arithmetic.

The first is the denominator. A probability is always a share of some group, and the question always names that group — "chosen at random from the whole survey", "chosen from among those under thirty", "chosen from the drama club". Five questions here work on nothing else. Each is built from a two-way breakdown written out in a sentence, and each carries a wrong option produced by using the whole total where only part of it applied. That wrong option is never nonsense: it is the correct answer to a different, clearly stated question, and the explanations say which one, because seeing what you did answer is the fastest way to notice what you were asked.

The second is what a study is allowed to claim, and it comes in two halves that are constantly confused. Random SELECTION — taking your people at random from a population — is what lets a result travel back to that population. Random ASSIGNMENT — splitting the people you have into groups at random — is what lets a difference be called a cause, because it makes the groups alike in everything except the one thing you changed. A study can have both, either, or neither, and each buys only its own kind of conclusion.

The set makes that concrete. One study assigns at random but takes its people from a single gym, so its causal claim is sound while its reach stops at people like those volunteers. Another surveys one private school in a wealthy neighborhood and announces a figure for a whole state, which fails not because a hundred people is too few — it is plenty for estimating a percentage — but because those hundred resemble one school and nowhere else.

Two questions concern a margin of error, and both carry the misreading that matters: treating an interval around a group's percentage as though it described every individual, or as though it described the sample itself. Within the sample the figure is known exactly; the margin is about everyone who was not asked.

Every study, school, town, company and researcher here is invented, no finding is attributed to any real organization, and nothing is drawn from any official publication.

• Read the denominator straight out of the question's wording, and restrict it when the question restricts the group
• Recognise a wrong answer as the correct answer to a different question, and name that question
• Use the complement rule as a check on a probability counted the long way
• Scale a sample result up to a population, and say what makes that legitimate
• Keep random selection and random assignment apart, and state which conclusion each one licenses
• Read a margin of error as an interval around the population's figure, not around the sample's or any individual's

Written for this catalogue, with no source document. The content is the probability and inference material named in the College Board's own public description of the Problem Solving and Data Analysis domain — probability and conditional probability from two-way data, inference from sample statistics, margin of error, and evaluating statistical claims; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test normally presents two-way data as a table on screen, which a text-only bank cannot reproduce: every breakdown here is written out in a sentence with all its counts named. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.