Understand that larger sample sizes generally reduce margin of error. | Inference from sample statistics and margin of error | SAT Math

Larger sample sizes generally produce smaller margins of error.

核心知识

Bigger samples give more precise estimates. As the sample size increases, the margin of error decreases.

深入理解

A sample of 50 people can swing noticeably depending on who happens to be included. A sample of 2,000 is far more stable — individual quirks get averaged out. This is why larger samples produce smaller margins of error.

Mathematically, the margin of error is proportional to $1 \sqrt 𝑛$ , where $𝑛$ is the sample size. Doubling the sample size doesn't cut the margin of error in half — it reduces it by a factor of $\sqrt 2 \approx 1.41$ . To actually halve the margin of error, you need to quadruple the sample size.

The SAT tests this relationship conceptually. You won't need to calculate exact margins, but you do need to know the direction: more data → narrower interval → more precise estimate.

分步讲解

Identify which sample is larger.
Recognize that the larger sample will have a smaller margin of error.
If comparing two studies on the same population, the one with the larger sample gives the more precise estimate.
To cut the margin of error in half, multiply the sample size by 4 (not 2).

常见误解

Thinking you need to double the sample size to halve the margin of error — you actually need to quadruple it because the relationship involves a square root.
Believing that a larger sample eliminates the margin of error entirely — it only reduces it, never to zero.
Confusing sample size with population size — what matters is how many you measure, not how large the population is.

题目

示例解析

Two researchers each conduct a random survey about the same population. Researcher A surveys 400 people and Researcher B surveys 1,600 people. How does Researcher B's margin of error compare to Researcher A's?

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