Question

a population has a mean (mu = 87) and a standard deviation (sigma = 29). find the mean and standard deviation of a sampling distribution of sample means with sample size (n = 235).
(mu_{overline{x}}=square) (simplify your answer.)
(sigma_{overline{x}}=square) (type an integer or decimal rounded to three decimal places as needed.)

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 $\mu = 87$, so $\mu_{\bar{x}}=87$.

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 $\sigma = 29$ and $n = 235$.
$$\sigma_{\bar{x}}=\frac{29}{\sqrt{235}}$$
First, calculate $\sqrt{235}\approx15.3297$.
Then, $\sigma_{\bar{x}}=\frac{29}{15.3297}\approx1.892$.

Answer:

$\mu_{\bar{x}} = 87$
$\sigma_{\bar{x}}\approx1.892$