**Problem: Decreasing the Margin of Error***Suppose you want to decrease the margin of error by a factor of 5 – for example, from 15% to 3%. By what factor must you increase the sample size?***Solution:**The sample size must be ___ times as great. *(Type a whole number.)*---**Explanation:**To decrease the margin of error by a factor, you need to understand the relationship between the margin of error and the sample size. The margin of error \( E \) is inversely proportional to the square root of the sample size \( n \), as given by the formula: \( E = \frac{Z \cdot \sigma}{\sqrt{n}} \).When you decrease the margin of error by a factor of \( k \), you need to increase the sample size by a factor of \( k^2 \).For example, to decrease the margin of error from 15% to 3%, which is a factor of \( \frac{15}{3} = 5 \), you increase the sample size by \( 5^2 = 25 \).Thus, to decrease the margin of error by a factor of 5, you must increase the sample size by a factor of 25.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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Suppose you want to decrease the margin of error by a factor of 5-for example, from 15% to 3%. By what factor must you increase the sample size? The sample size must be times as great. (Type a whole number.)