Question

7.49 Mobile Homes. According to the U.S. Census Bureau publication Manufactured Housing Statistics, the mean price of new mobile homes is $\backslash (65,100$. Assume a standard deviation of $\backslash )7200$. Let $\tilde{x}$ denote the mean price of a sample of new mobile homes.
a. For samples of size $50$, find the mean and standard deviation of $\overline{x}$. Interpret your results in words.
b. Repeat part (a) with $n=100$.

Short answer

(a) The mean is $\$65,100$and the standard deviation is$\$1018.2$.

(b) The mean is $\$65,100$ and the standard deviation is $\$720$.

Step-by-step solution

Part (a) Step 1: Given information

To find the mean and the standard deviation for the sample size of $50$. Where, the mean price of new mobile homes is $\$65,100$and the standard deviation of $\$7200$.

Part (a) Step 2: Explanation

The mean $\overline{x}$indicates the price of a sample of new mobile homes.

The mean price of new mobile homes is $\$65,100$. And the standard deviation is $\$7200$.

So, $\mu =\$65,100$and $\sigma =\$7200$.
Then the Size of the sample,$n=50$.
Mean of $\overline{x}$,
${\mu }_{\overline{x}}=\mu$

${\mu }_{\overline{x}}=\$65,100$

Standard deviation of $\overline{x}$,

$\begin{array}{r}{\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\end{array}$
$=\frac{7200}{\sqrt{50}}$
$=\frac{7200}{7.071068}$
$=1018.2$
The mean and standard deviation of all possible samples mean prices for $50$ new mobile homes are $\$65,100$and $\$1018.2$ respectively.

Part (b) Step 1: Given information

To find the mean and the standard deviation for the sample size of $100$. Where, the mean price of new mobile homes is $\$65,100$and the standard deviation of $\$7200$.

Part (b) Step 2: Explanation

Standard deviation of $\overline{x}$is $\begin{array}{r}{\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\end{array}$

Where the sample size is $n=100$.
Mean of $\overline{x}$is:

${\mu }_{\overline{x}}=\mu$
${\mu }_{\overline{x}}=65,100$
Then,

$\begin{array}{r}{\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\end{array}$
$=\frac{7200}{\sqrt{100}}$
$=\frac{7200}{10}$
$=720$
The mean and standard deviation of all possible samples mean prices for $100$new mobile homes are$\$65,100$and $\$720$, respectively.