Question

ch. 15 homework (due on 10/9)
a1. the mean height of a female in the united states is approximately normal with mean 65 inches and standard deviation 2.9 inches. you choose an srs of 50 females and average their heights. if you take many different samples of 50 females, the mean, $mu_{\bar{x}}$, of the average heights will be close to ____________ and the standard deviation, $sigma_{\bar{x}}$, of the average heights will be close to ____________.

Explanation:

Step1: Recall the mean of the sampling - distribution of the sample mean

The mean of the sampling - distribution of the sample mean $\mu_{\bar{X}}$ is equal to the population mean $\mu$. Given that the population mean $\mu = 65$ inches, for samples of size $n = 50$, $\mu_{\bar{X}}=\mu$.
$\mu_{\bar{X}} = 65$

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

The standard deviation of the sampling - distribution of the sample mean (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, $\sigma = 2.9$ inches and $n = 50$.
$\sigma_{\bar{X}}=\frac{2.9}{\sqrt{50}}\approx\frac{2.9}{7.071}\approx0.41$

Answer:

The mean, $\mu_{\bar{X}}$, of the average heights will be close to $65$ and the standard deviation, $\sigma_{\bar{X}}$, of the average heights will be close to $0.41$.