In the context of the sampling distribution of the sample mean, what is the standard error of the mean $\overline{X}$ for a population with standard deviation $\sigma$ and sample size $n$?

• A. $\frac{\sigma }{\sqrt{n}}$
• B. $\frac{\sigma }{n}$
• C. $\frac{\sqrt{\sigma }}{n}$
• D. $\sigma \surd n$

Answer: (A)

Verified step by step guidance

1. Recall that the standard error of the mean (SEM) measures the variability of the sample mean \( \overline{X} \) from the population mean when taking samples of size \( n \).
2. Understand that the population standard deviation is denoted by \( \sigma \), and the sample size is \( n \).
3. The formula for the standard error of the mean is derived from the population standard deviation divided by the square root of the sample size, which accounts for the fact that larger samples tend to have less variability in their means.
4. Write the formula for the standard error of the mean as: \[ \\text{SEM} = \frac{\\sigma}{\\sqrt{n}} \]
5. This formula shows that as the sample size \( n \) increases, the standard error decreases, reflecting more precise estimates of the population mean.