Question

a population has a mean $mu = 151$ and a standard deviation $sigma = 20$. find the mean and standard deviation of the sampling distribution of sample means with sample size $n = 60$.
the mean is $mu_{\bar{x}}=square$, and the standard deviation is $sigma_{\bar{x}}=square$. (round to three decimal places as needed.)

Explanation:

Step1: Recall the mean of sampling distribution

The mean of the sampling distribution of sample means $\mu_{\bar{x}}$ is equal to the population mean $\mu$.
$\mu_{\bar{x}}=\mu$

Step2: Substitute the population - mean value

Given $\mu = 151$, so $\mu_{\bar{x}}=151$.

Step3: Recall the formula for the standard deviation of sampling distribution

The standard deviation of the sampling distribution of sample means (also known as the standard error) is given by $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$

Step4: Substitute the given values

Given $\sigma = 20$ and $n = 60$. Then $\sigma_{\bar{x}}=\frac{20}{\sqrt{60}}$.
$\sigma_{\bar{x}}=\frac{20}{\sqrt{60}}\approx\frac{20}{7.746}\approx2.582$

Answer:

The mean is $\mu_{\bar{x}} = 151$, and the standard deviation is $\sigma_{\bar{x}}\approx2.582$.