Answer to Question #170466 in Statistics and Probability for Melissa M Elliott

Question

A contractor decided to build homes that will include the middle 80% of the market. If the average size of homes built is 1750 square feet, find the maximum and minimum sizes of the homes the contractor should build. Assume that the standard deviation is 96 square feet and the variable is normally distributed.

Group of answer choices

Answer 1

A contractor decided to build homes that will include the middle 80% of the market. If the average size of homes built is 1750 square feet, find the maximum and minimum sizes of the homes the contractor should build. Assume that the standard deviation is 96 square feet and the variable is normally distributed.

We have that:

$\mu=1750$

$\sigma=96$

Let a and b denote minimum and maximum sizes of the homes respectively.

Thus $P(a<X<b)=80\%=0.8$

$P(X<b)=90\%=0.9$ and $P(X<a)=10\%=0.1$

$P(X<b)=P(Z<\frac{b-\mu}{\sigma})=0.9\implies \frac{b-1750}{96}=1.28\implies b=1873$

$P(X<a)=P(Z<\frac{a-\mu}{\sigma})=0.1\implies \frac{a-1750}{96}=-1.28\implies a=1627$

Answer: the minimum size is 1627, the maximum size is 1873