Explain why quadrupling the sample size causes the margin of error to be cut in half.

Verified step by step guidance

1. Recall that the margin of error (ME) in estimating a population parameter using a sample is often given by the formula: \(\text{ME} = z^* \times \frac{\sigma}{\sqrt{n}}\), where \(z^*\) is the critical value, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
2. Notice that the margin of error is inversely proportional to the square root of the sample size, \(\sqrt{n}\). This means that as the sample size increases, the margin of error decreases, but not linearly—rather, it decreases with the square root of \(n\).
3. If we quadruple the sample size, the new sample size becomes \$4n\(. Substitute this into the denominator under the square root: \)\sqrt{4n} = 2\sqrt{n}$.
4. Because the denominator doubles (from \(\sqrt{n}\) to \$2\sqrt{n}\(), the entire fraction \)\frac{\sigma}{\sqrt{n}}$ is cut in half. Since the margin of error is directly proportional to this fraction, the margin of error is also cut in half.
5. Therefore, quadrupling the sample size reduces the margin of error by a factor of \(\frac{1}{2}\), explaining why the margin of error is halved.

Key Concepts

Margin of Error

The margin of error quantifies the range within which the true population parameter is expected to lie, based on a sample statistic. It reflects the uncertainty in estimates and is influenced by sample size, variability, and confidence level.

Sample Size and Its Effect on Variability

Increasing the sample size reduces the variability of the sample mean or proportion, leading to more precise estimates. Larger samples provide more information, which decreases the standard error and thus narrows the margin of error.

Relationship Between Sample Size and Margin of Error

The margin of error is inversely proportional to the square root of the sample size. Quadrupling the sample size increases the denominator by a factor of two (since √4=2), which cuts the margin of error in half, improving estimate precision.