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?

Short 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.

Step-by-step solution

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.