Assuming a $90\%$ confidence level for a population mean with known standard deviation, the margin of error is approximately equal to the critical value times the standard error. Which of the following best represents the margin of error formula for this scenario?

• A. $1.28\times \frac{\sigma }{\sqrt{n}}$
• B. $2.58\times \frac{\sigma }{\sqrt{n}}$
• C. $1.645\times \frac{\sigma }{\sqrt{n}}$
• D. $1.96\times \frac{\sigma }{\sqrt{n}}$

Answer: (C)

Verified step by step guidance

1. Identify the confidence level given in the problem, which is 90%. This confidence level determines the critical value (z-score) from the standard normal distribution.
2. Recall that the margin of error (ME) formula for a population mean with known standard deviation is given by: \[ ME = z^* \times \frac{\sigma}{\sqrt{n}} \] where \(z^*\) is the critical value, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
3. Find the critical value \(z^*\) corresponding to a 90% confidence level. This value is the z-score that captures the middle 90% of the standard normal distribution, leaving 5% in each tail.
4. Substitute the critical value \(z^*\), the population standard deviation \(\sigma\), and the sample size \(n\) into the margin of error formula.
5. Compare the given options with the formula you have constructed to identify which one correctly represents the margin of error for a 90% confidence level.