Question

chapter 6 quiz - continuous probability distributions 10 points possible answered: 8/11 question 11 a population of values has an unknown distribution with $mu = 207.8$ and $sigma = 63.4$. you intend to draw a random sample of size $n = 55$. 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 the sampling - distribution of sample means

The mean of the distribution of sample means $\mu_{\bar{x}}$ is equal to the population mean $\mu$. Given $\mu = 207.8$, so $\mu_{\bar{x}}=\mu$.
$\mu_{\bar{x}} = 207.8$

Step2: Recall the formula for the standard deviation of the sampling - distribution of sample means

The standard deviation of the distribution of sample means (also known as 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 $\sigma = 63.4$ and $n = 55$.
$\sigma_{\bar{x}}=\frac{63.4}{\sqrt{55}}\approx\frac{63.4}{7.42}\approx8.55$

Answer:

$\mu_{\bar{x}} = 207.8$
$\sigma_{\bar{x}}\approx8.55$