Question

a population has a mean $\mu = 72$ and a standard deviation $\sigma = 6$. find the mean and standard deviation of a sampling distribution of sample means with sample size $n = 36$.
$\mu_{\overline{x}}=\square$ (simplify your answer.)
$\sigma_{\overline{x}}=\square$ (simplify your answer.)

Explanation:

Step1: Find the mean of the sampling distribution of sample means

The mean of the sampling distribution of sample means ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$).
$$\mu_{\bar{x}}=\mu$$
Given $\mu = 72$, so $\mu_{\bar{x}}=72$.

Step2: Find the standard deviation of the sampling distribution of sample means

The standard deviation of the sampling distribution of sample means ($\sigma_{\bar{x}}$) 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 $\sigma = 6$ and $n = 36$.
$$\sigma_{\bar{x}}=\frac{6}{\sqrt{36}}=\frac{6}{6}=1$$

Answer:

$\mu_{\bar{x}} = 72$; $\sigma_{\bar{x}}=1$