Question

according to a recent publication, the mean price of new mobile homes is $60,300. assume a standard deviat $7900. 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 $60300 and $1580.00, respectively.
(round to the nearest cent as needed.)
b. for the 50 mobile homes sampled, 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 mean

The mean of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\). Given \(\mu = 60300\), for any sample size \(n\), \(\mu_{\bar{x}}=\mu\).

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

The formula for the standard deviation of the sample mean (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Here \(\sigma = 7900\).

For \(n = 50\):

• Mean of \(\bar{x}\):

Since \(\mu_{\bar{x}}=\mu\), and \(\mu = 60300\), so \(\mu_{\bar{x}}=60300\).

• Standard deviation of \(\bar{x}\):

Using the formula \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), substitute \(\sigma = 7900\) and \(n = 50\).

$$ \sigma_{\bar{x}}=\frac{7900}{\sqrt{50}}\approx\frac{7900}{7.071}\approx1117.24 $$

Answer:

For part (b), the mean is \(\$60300\) and the standard deviation is \(\$1117.24\)