Q. 22 - Elementary Statistics

Question

Prices of New Mobile Homes. The U.S. Census Bureau publishes annual price figures for new mobile homes in Manufactured Housing Statistics. The prices of a sample of $250$ new mobile homes have roughly a bell-shaped distribution with mean $63.3$  thousand and standard deviation $7.9$ thousand.
a. Fill in the blanks: Approximately $95\%$ of the mobile homes in the sample have prices between ___ thousand and ___ thousand.
b. Fill in the blank: Approximately
____% of the mobile homes in the sample have prices between $39.6$ thousand and $87.0$ thousand.
c. Approximately how many mobile homes in the sample have prices between $55.4$ thousand and  $71.2$ thousand?

Step-by-Step Solution

Answer

Part(a) Approximately $95\%$ of the mobile homes in the sample have prices between  $47.5$ thousand and $79.1$ thousand.

Part(b) $99.7\%$ of the mobile homes in the sample have prices between $39.6$ thousand and $87.0$ thousand.

Part(c) $170$  of the mobile homes in the sample have prices between $55.4$ thousand and $71.2$  thousand.

Part(a) Step 1: Given information

We are given sample of prices of mobile homes in manufactured housing statistics of U.S. Census Bureau.

Part(a) Step 2: Simplify

According to empirical rule ,

For $95\%$ of the sample there should be two standard deviations at either side of mean, therefore,

$$\begin{gathered} \overline{x}-2s=63.3-2\left(7.9\right)=47.5 \\ \overline{x}+2s=63.3+2\left(7.9\right)=79.1 \end{gathered}$$

Approximately $95\%$ of the mobile homes in the sample have prices between  $47.5$ thousand and $79.1$ thousand.

Part(b) Step 1: Given information

We are given sample of prices of mobile homes in manufactured housing statistics of U.S. Census Bureau.

Part(b) Step 2: Simplify

According to empirical rule,

$$\begin{gathered} \overline{x}-ks=39.6 \\ 63.3-k\left(7.9\right)=39.6 \\ k=3 \\ \overline{x}+ks=80.70 \\ 63.3+k\left(7.9\right)=87.0 \\ k=3 \end{gathered}$$

 Now, as $k=3$ that means $99.7\%$ of mobiles in the sample lies in given range.

$250\times \frac{99.7}{100}=249.25\approx 249$

$99.7\%$ of the mobile homes in the sample have prices between $39.6$ thousand and  $87.0$ thousand.

Part(c) Step 1: Given information

We are given sample of prices of mobile homes in manufactured housing statistics of U.S. Census Bureau.

Part(c) Step 2: Simplify

According to empirical rule,

$$\begin{gathered} \overline{x}-ks=55.4 \\ 63.3-k\left(7.9\right)=55.4 \\ k=1 \\ \overline{x}+ks=71.2 \\ 63.3+k\left(7.9\right)=71.2 \\ k=1 \end{gathered}$$

 Now, as  that means $68\%$ of mobiles in the sample lies in given range.

$250\times \frac{68}{100}=170$

$170$ of the mobile homes in the sample have prices between $55.4$ thousand and  $71.2$ thousand.