Question

1. At a particular amusement park, most of the live characters have height requirements of a minimum of 57 in. and a maximum of 63 in. A survey found that​ women's heights are normally distributed with a mean of 62.4 in. and a standard deviation of 3.6 in. The survey also found that​ men's heights are normally distributed with a mean of 68.3 in. and a standard deviation of 3.6 in.
   
   Part 1:
   Find the percentage of men meeting the height requirement.
   
   The percentage of men who meet the height requirement is ____?____.
   ​(Round answer to nearest hundredth of a percent - i.e. 23.34%)
   
   What does the result suggest about the genders of the people who are employed as characters at the amusement​ park?
   Since most men___?___ the height​ requirement, it likely that most of the characters are ___?___ .
   (Use "meet" or "do not meet" for the first blank and "men" or "women" for the second blank.)
   
   Part 2: I was able to solve part 2 on my own.
   If the height requirements are changed to exclude only the tallest​ 50% of men and the shortest​ 5% of​ men, what are the new height​ requirements?
   The new height requirements are a minimum of 62.4 in. and a maximum of 68.3 in.
   ​(Round to one decimal place as​ needed.)

Answer 1

Part 1:

(i)

For Men:

= 68.3

= 3.6

To find P(57 < X < 63):

Case 1: For X from 57 to mid value:
Z = (57 - 68.3)/3.6

= - 3.1389

Table of Area Under Standard Normal Curve gives area = 0.4992

Case 2: For X from 63 to mid value:
Z = (63 - 68.3)/3.6

= - 1.4722

Table of Area Under Standard Normal Curve gives area = 0.4292

So,

P(57 < X < 63) = 0.4992 - 0.4292 = 0.07 = 7.00 %

So,

The percentage of men who meet the height requirement is 7.00 %

(ii)

Since most men do not the height​ requirement, it likely that most of the characters are women.

Part 2:

Smallest 5% corresponds to area = 0.50 - 0.05 = 0.45 from mid value to Z on LHS.

Table of Area Under Standard Normal Curve gives Z = - 1.645

So,

Z = - 1.645 = (X - 68.3)/3.6

So,

X = 68.3 - (1.645 X 3.6) = 68.3 - 5.922 = 62.4

Maximum = 68.3 since tallest 50%

So,

Answer is:

The new height requirements are a minimum of 62.4 in. and a maximum of 68.3 in