Q. 7.49 - Elementary Statistics

Question

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 x denoted the mean price of a sample of new mobile homes.

Part (a): For samples of size $50$, find the mean and standard deviation of $\overline{x}$. Interpret your results in words.

Part (b): Repeat part (a) with $n=100$.

Step-by-Step Solution

Answer

Part (a): The mean and standard deviation of $\overline{x}$ are $\$65,100$ and $\$1018.20$.

On interpreting, for samples of 50 new mobile homes, the mean and standard deviation of all possible sample mean prices are $\$65,100$ and $\$1018.20$ respectively.

Part (b): The mean and standard deviation of $\overline{x}$ are $\$65,100$ and $\$720$.

On interpreting, for samples of 50 new mobile homes, the mean and standard deviation of all possible sample mean prices are $\$65,100$ and $\$720$.

Part (a) Step 1. Given information.

Consider the given question,

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

The sample size is 50.

Part (a) Step 2. Find the mean and standard deviation when n = 50 .

The mean of $\overline{x}$,

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

Standard deviation of $\overline{x}$,

$$\begin{gathered} {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}} \\ =\frac{7200}{\sqrt{50}} \\ =\frac{7200}{7.071068} \\ =\$1018.20 \end{gathered}$$

On interpreting, we can say that for samples of 50 new mobile homes, the mean and standard deviation of all possible sample mean prices are $\$65,100$ and  $\$1018.20$.

Part (b) Step 1. Find the mean and standard deviation when n = 100 .

The mean of $\overline{x}$,

$$\begin{gathered} {\mu }_{\overline{x}}=\mu \\ =\$65,100 \end{gathered}$$
Standard deviation of $\overline{x}$,

$$\begin{gathered} {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}} \\ =\frac{7200}{\sqrt{100}} \\ =\frac{7200}{10} \\ =\$720 \end{gathered}$$

On interpreting, we can say that for samples of 100 new mobile homes, the mean and standard deviation of all possible sample mean prices are $\$65,100$ and $\$720$.