Which of the following calculations is not derived from the confidence interval for a population mean $\mu$?

• A. The lower and upper bounds of the interval
• B. The sample mean $\mathrm{\&xmacr;}$
• C. The population variance ${\sigma }^2$
• D. The margin of error $E$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand what a confidence interval for a population mean represents. It is an interval estimate that provides a range of plausible values for the population mean based on the sample data.
2. Step 2: Recall the formula for a confidence interval for a population mean when the population variance is unknown: \(\bar{x} \pm E\), where \(\bar{x}\) is the sample mean and \(E\) is the margin of error.
3. Step 3: Identify the components derived from the confidence interval: the lower bound (\(\bar{x} - E\)), the upper bound (\(\bar{x} + E\)), the sample mean (\(\bar{x}\)), and the margin of error (\(E\)).
4. Step 4: Recognize that the population variance is a parameter of the population and is not directly calculated from the confidence interval; instead, it is often unknown and estimated by the sample variance.
5. Step 5: Conclude that among the options, the population variance is not derived from the confidence interval for a population mean, while the other quantities are directly involved in or derived from the confidence interval calculation.