Question

a population has mean \\( \mu = 25 \\) and standard deviation \\( \sigma = 3 \\). find \\( \mu_{\bar{x}} \\) and \\( \sigma_{\bar{x}} \\) for samples of size \\( n = 25 \\). round your answers to one decimal place if needed.

Explanation:

Step1: Find the mean of the sample mean ($\mu_{\bar{x}}$)

The mean of the sampling distribution of the sample mean ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$). Given $\mu = 25$, so $\mu_{\bar{x}}=\mu = 25$.

Step2: Find the standard deviation of the sample mean ($\sigma_{\bar{x}}$)

The standard deviation of the sampling distribution of the sample mean (also called the standard error) is given by the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
Given $\sigma = 3$ and $n = 25$, we calculate:
$\sigma_{\bar{x}}=\frac{3}{\sqrt{25}}=\frac{3}{5}=0.6$

Answer:

$\mu_{\bar{x}} = 25$, $\sigma_{\bar{x}} = 0.6$