Question

A survey found that​ women's heights are normally distributed with mean 63.7 in. and standard deviation 2.4 in. The survey also found that​ men's heights are normally distributed with mean 67.4 in. and standard deviation 3.7 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 57 in. and a maximum of 63 in. Complete parts​ (a) and​ (b) below. a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement​ park?

Answer 1

The percentage of men meeting the height requierements is 11.47%.

As women have a less mean height, closer to the height requierments than men's mean height, it is expected that more percentage of women will fall within the height requirements.

Explanation:

We have men's height normally distributed with mean 67.4 in and standard deviation 3.7 in.

The amusement park requires heights between 57 in and 63 in.

We can calculate the proportion using the z-scores and the standard normal probability distribution.

`z_1=(X_1-\mu)/(\sigma)=(57-67.4)/(3.7)=(-10.4)/(3.7)=-2.811\\\\\\z=(X-\mu)/(\sigma)=(63-67.4)/(3.7)=(-4.4)/(3.7)=-1.189`

`P(57<X<63)=P(-2.811<z<-1.189)\\\\P(57<X<63)=P(z<-1.189)-P(z<-2.811)\\\\P(57<X<63)=0.1172-0.0025\\\\P(57<X<63)=0.1147`

The percentage of men meeting the height requierements is 11.47%.