To calculate the margin of error for a confidence interval, you can use the formula:
\[ \text{Margin of Error} = z \times \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( z \) is the z-score corresponding to the confidence level,
- \( \sigma \) is the standard deviation,
- \( n \) is the sample size.
For a 90% confidence level, the z-score (which corresponds to the critical value for a two-tailed test) is approximately 1.645.
Given:
- Standard deviation (\( \sigma \)) = 5
- Sample size (\( n \)) = 500
Now, let's calculate the standard error (SE):
\[ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{500}} \]
Calculating \( \sqrt{500} \):
\[ \sqrt{500} \approx 22.36 \]
Now calculate the standard error:
\[ \text{Standard Error} = \frac{5}{22.36} \approx 0.2236 \]
Now, compute the margin of error:
\[ \text{Margin of Error} = 1.645 \times 0.2236 \approx 0.3674 \]
Finally, rounding the margin of error to the nearest hundredth:
\[ \text{Margin of Error} \approx 0.37 \]
So, the margin of error is approximately 0.37.