Effect of quadrupling the sample on the margin

Question

You increase a sample from n = 200 to n = 800, keeping the population spread and confidence level fixed. By what factor does the margin of error change?

Accepted Answer

Setup. The margin of error is proportional to 1/\sqrt{n}, so the change factor is \sqrt{\frac{n{	ext{old}}}{n{	ext{new}}}} = \sqrt{\frac{200}{800}}. Compute. \sqrt{\frac{200}{800}} = \sqrt{\frac{1}{4}} = \frac{1}{2}. The margin is multiplied by 0.5; it halves. Interpret. Quadrupling the sample halves the margin. This is the same square-root law seen from the other direction. Takeaway. Multiply the sample by 4 and the margin drops to half. The improvement factor is always the square root of the sample-size ratio.

Effect of quadrupling the sample on the margin

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