Given that:
[tex]\begin{gathered} n=500 \\ x=100 \end{gathered}[/tex]
You need to use the following formula in order to calculate the Margin of error E:
[tex]E=Z_{\frac{a}{2}}\sqrt[]{\frac{pq}{n}}[/tex]
Where "p" is the probability of success, "q" is the probability of failure, "Z" is z-score, and "n" is the sample size.
In this case, since you need to use a 95% degree of confidence, by definition:
[tex]Z_{\frac{a}{2}}=1.96[/tex]
By definition:
[tex]\begin{gathered} p=\frac{x}{n} \\ \\ q=1-p \end{gathered}[/tex]
Substituting the values of "n" and "x" into the first formula and evaluating, you get that:
[tex]p=\frac{100}{500}=0.2[/tex]
Therefore, "q" is:
[tex]\begin{gathered} q=1-0.2 \\ q=0.8 \end{gathered}[/tex]
Knowing all those values, you can substitute them into the formula for calculating the Margin of Error:
[tex]E=1.96\sqrt[]{\frac{(0.2)(0.8)}{500}}[/tex]
Finally, evaluating, you get:
[tex]\begin{gathered} E=1.96\sqrt[]{\frac{0.16}{500}} \\ \\ E\approx0.0351 \end{gathered}[/tex]
Therefore, the answer is:
[tex]E\approx0.0351[/tex]
