Question

according to a recent publication, the mean price of new mobile homes is $60,300. assume a standard deviation of $7100. let \\( \overline { 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 \\( \overline { x } \\). interpret your results in words
b. repeat part (a) with \\( n = 50 \\).
a. for the mean and standard deviation of 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 $\bar{x}$, denoted as $\mu_{\bar{x}}$, is equal to the population mean $\mu$.
Given $\mu = 60300$, so $\mu_{\bar{x}}=\mu = 60300$.

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

The standard deviation 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.
For $n = 25$ and $\sigma=7100$, we have $\sigma_{\bar{x}}=\frac{7100}{\sqrt{25}}=\frac{7100}{5}=1420$.

Step3: Interpretation for part (a)

The mean of the sample mean $\bar{x}$ (for samples of size $n = 25$) is the same as the population mean. This means that, on average, the sample mean price of new mobile homes (for samples of size 25) is equal to the population mean price of new mobile homes. The standard deviation of the sample mean (standard error) of 1420 indicates the amount of variation or dispersion of the sample mean values around the population mean.

Step4: Repeat for $n = 50$

For the mean of the sample mean, $\mu_{\bar{x}}=\mu=60300$ (still holds as it is a property of the sample - mean distribution, $\mu_{\bar{x}}$ does not depend on $n$).
For the standard deviation of the sample mean, $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, with $n = 50$ and $\sigma = 7100$, we have $\sigma_{\bar{x}}=\frac{7100}{\sqrt{50}}\approx\frac{7100}{7.071}\approx1004.10$.

Step5: Interpretation for part (b)

The mean of the sample mean $\bar{x}$ (for samples of size $n = 50$) is equal to the population mean. So, on average, the sample mean price of new mobile homes (for samples of size 50) is the same as the population mean price of new mobile homes. The standard deviation of the sample mean (standard error) of approximately 1004.10 (which is less than the standard error for $n = 25$) indicates that as the sample size increases, the sample mean values are less dispersed around the population mean.

Answer:

a. For samples of size 25, the mean and standard deviation of $\bar{x}$ are $\$60300$ and $\$1420$, respectively.
b. For samples of size 50, the mean and standard deviation of $\bar{x}$ are $\$60300$ and $\$1004.10$, respectively.