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.

1. 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.

1. 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: Calculate the mean of the sample mean ($\bar{x}$)

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

Step2: Calculate the standard deviation of the sample mean ($\sigma_{\bar{x}}$) for \(n = 25\)

The formula for the standard deviation of the sample mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 8000$ and $n = 25$.
$\sigma_{\bar{x}}=\frac{8000}{\sqrt{25}}=\frac{8000}{5}=1600$.

Step3: Calculate the standard deviation of the sample mean ($\sigma_{\bar{x}}$) for \(n = 50\)

Using the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, with $\sigma = 8000$ and $n = 50$.
$\sigma_{\bar{x}}=\frac{8000}{\sqrt{50}}\approx\frac{8000}{7.071}\approx1131$.

Answer:

a. The mean of all possible sample mean prices is $\$65300$ and the standard deviation is $\$1600$.
b. When \(n = 50\), the mean of the sample mean ($\bar{x}$) is $\mu_{\bar{x}} = 65300$ (which means on average, the sample mean price of a sample of new mobile homes will be equal to the population mean price of new mobile homes). The standard deviation $\sigma_{\bar{x}}\approx1131$, which is smaller than when \(n = 25\). As the sample size \(n\) increases, the spread of the sample means (standard deviation of \(\bar{x}\)) decreases.