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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.)

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.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**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.
Transcribed Image Text:**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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