Question

Suppose heights of adult females follow a normal distribution with a mean of 65 inches and a standard deviation of 3.5 inches. A modeling agency requires female fashion models to be at least 5 feet and 8 inches tall (i.e., 68 inches tall). What is the probability of randomly selecting an adult female whose height is greater than the 68 inch requirement for fashion models at this agency? To show your work, draw a sketch of the probability distribution, shade the correct area and provide a probability statement. Round your answer to 4 decimal places.

Answer 1

Answer with explanation:

Given : The heights of adult females follow a normal distribution with a mean of 65 inches and a standard deviation of 3.5 inches.

i.e. $\mu=65$    $\sigma=3.5$

A modeling agency requires female fashion models to be at least 5 feet and 8 inches tall (i.e., 68 inches tall).

Let x denotes the height of adult females.

Formula = $z=\dfrac{x-\mu}{\sigma}$

Then, the probability of randomly selecting an adult female whose height is greater than the 68 inch requirement for fashion models at this agency :

$P(x>68)=P(\dfrac{x-\mu}{\sigma}>\dfrac{68-65}{3.5})[\\\\\approx P(z>0.86)\ \ [\text{Rounded to two-decimal places.}]\\\\=1-P(z<0.86)\ \ [\because P(Z>z)=1-P(Z<z)]\\\\=1-0.8051=0.1949\ \ [\text{Using the z-table}]$

The shape of normal curve is bell -shaped.

Sketch of the probability distribution is attached below.

Hence, the required probability = 0.1949