What quadrupling the sample does to the margin

Question

A 95% confidence interval for a mean has a margin of error (half-width) of $4$ when the sample size is $n = 100$ . You increase the sample to $n = 400$ , keeping the same population spread and confidence level.

What is the new margin of error?

Accepted Answer

Setup. The margin of error is z^{}\sigma/\sqrt{n}. With z^{} and \sigma fixed, the margin scales as 1/\sqrt{n}. Compute. Going from n = 100 to n = 400 multiplies n by 4, so \sqrt{n} doubles and the margin is divided by 2: 	ext{new margin} = 4 	imes \sqrt{\frac{100}{400}} = 4 	imes \frac{1}{2} = 2. Interpret. Quadrupling the data only halves the margin. Precision improves slowly because it depends on the square root of sample size. Takeaway. Margin of error shrinks like 1/\sqrt{n}. To cut it in half you need four times the data, not twice.

What quadrupling the sample does to the margin

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