Question

according to a recent publication, the mean price of new mobile homes is $65,300. assume a standard deviation of $8000. let x denote the mean price of a sample of new mobile homes.
a. for samples of size 25, find the mean and standard deviation of x. interpret your results in words.
b. repeat part (a) with n = 50.
a. for samples of 25 mobile homes, the mean and standard deviation of all possible sample mean prices are $□ and $□, respectively.
(round to the nearest cent as needed.)

Explanation:

Step1: Recall the formula for the mean of the sample means

The mean of the sample means ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$). Given $\mu = 65300$.

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

The formula for the standard deviation of the sample means (also known as the standard error) is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$.

For part (a) where $n = 25$ and $\sigma=8000$

• Calculate the standard deviation of the sample means:

$\sigma_{\bar{x}}=\frac{8000}{\sqrt{25}}=\frac{8000}{5}=1600$

For part (b) where $n = 50$ and $\sigma = 8000$

• Calculate the standard deviation of the sample means:

$\sigma_{\bar{x}}=\frac{8000}{\sqrt{50}}\approx\frac{8000}{7.071}\approx1131$

Answer:

a. The mean of all possible sample - mean prices is $\$65300$ and the standard deviation is $\$1600$.
b. The mean of all possible sample - mean prices is $\$65300$ (since $\mu_{\bar{x}}=\mu$ always) and the standard deviation is approximately $\$1131$.