Question

find the margin of error for a survey that has the given sample size of 500 people. use this formula: ±\frac{1}{\sqrt{n}}. round to the nearest thousandth.

Explanation:

Step1: Identify the sample size

The sample size \( n = 500 \).

Step2: Calculate the square root of the sample size

First, find \( \sqrt{n}=\sqrt{500} \). We know that \( \sqrt{500}=\sqrt{100\times5} = 10\sqrt{5}\approx10\times2.23607 = 22.3607 \).

Step3: Calculate the margin of error

The formula for the margin of error is \( \pm\frac{1}{\sqrt{n}} \). Substitute \( \sqrt{n}\approx22.3607 \) into the formula: \( \frac{1}{22.3607}\approx0.0447 \). Rounding to the nearest thousandth (three decimal places), we look at the fourth decimal place. The number is \( 0.0447\), the fourth decimal is \( 7\), which is greater than or equal to \( 5 \), so we round up the third decimal place. \( 0.0447\approx0.045 \) (and also \( - 0.045 \) for the negative part).

Answer:

\( \pm0.045 \)