Question

a population of values has an unknown distribution with $mu = 230.3$ and $sigma = 35.3$. you intend to draw a random sample of size $n = 174$. what is the mean of the distribution of sample means? $mu_{\bar{x}}=$ (please enter an exact answer.) what is the standard deviation of the distribution of sample means? $sigma_{\bar{x}}=$ (please report your answer accurate to 2 decimal places.)

Explanation:

Step1: Recall the mean of sample - means formula

The mean of the distribution of sample means $\mu_{\bar{x}}$ is equal to the population mean $\mu$. Given $\mu = 230.3$, so $\mu_{\bar{x}}=\mu$.

Step2: Recall the standard - deviation of sample - means formula

The standard deviation of the 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. Here, $\sigma = 35.3$ and $n = 174$. So, $\sigma_{\bar{x}}=\frac{35.3}{\sqrt{174}}$.
First, calculate $\sqrt{174}\approx13.19$. Then, $\sigma_{\bar{x}}=\frac{35.3}{13.19}\approx2.68$.

Answer:

$\mu_{\bar{x}} = 230.3$
$\sigma_{\bar{x}}\approx2.68$