Question

according to a recent publication, the mean price of new mobile homes is $64,500. assume a standard deviation of $6600. 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.

Explanation:

Step1: <Mean of sample mean>

The mean of the sample mean \(\bar{x}\), \(\mu_{\bar{x}}=\mu\). Given \(\mu = 64500\), so \(\mu_{\bar{x}}=64500\).

Step2: <Standard deviation of sample mean (for \(n = 25\))>

The formula for the standard deviation of the sample mean is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 6600\) and \(n = 25\), then \(\sigma_{\bar{x}}=\frac{6600}{\sqrt{25}}=\frac{6600}{5}=1320\).
Interpretation: The mean of the sample means of samples of size \(n = 25\) is the same as the population mean. The standard deviation of the sample means (standard error) represents the average amount that the sample mean \(\bar{x}\) deviates from the population mean \(\mu\).

Step3: <Standard deviation of sample mean (for \(n = 50\))>

Using the formula \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\) with \(n = 50\) and \(\sigma=6600\), we have \(\sigma_{\bar{x}}=\frac{6600}{\sqrt{50}}\approx\frac{6600}{7.071}\approx933.4\)

Answer:

a. Mean of \(\bar{x}\): \(\$64500\), Standard deviation of \(\bar{x}\): \(\$1320\). Interpretation: The average value of the sample means of samples of size \(25\) is the population mean. The sample - mean values deviate from the population mean by an average of \(\$1320\).
b. Mean of \(\bar{x}\): \(\$64500\), Standard deviation of \(\bar{x}\): \(\approx\$933.4\)