Question

1. The heights of a large population of college students are approximately normally distributed with mean 5 feet 6 inches and standard deviation 8 inches.
   1. What fraction of the students are shorter than 6 feet 6 inches?
   2. If there are approximately 38,000 students at the university, approximately how many are shorter than 5 feet?
   3. A student wants to start a club for the tallest students on campus. They decide that membership requires that a person be in the top 3% of height for the university. What is the minimum height requirement?

Answer 1

Use conversion 1 feet = 12 inch.

a)

X ~ N ( µ = 66 , σ = 8 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 78 ) = P ( Z < 78 - 66 ) / 8 )
= P ( Z < 1.5 )
P ( X < 78 ) = 0.9332

b)

X ~ N ( µ = 66 , σ = 8 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 60 ) = P ( Z < 60 - 66 ) / 8 )
= P ( Z < -0.75 )
P ( X < 60 ) = 0.2266

Of the 38000, we expect, 8611 students are shorter than 5 feet

c)

X ~ N ( µ = 66 , σ = 8 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.03 = 0.97
To find the value of x
Looking for the probability 0.97 in standard normal table to calculate critical value Z = 1.8808
Z = ( X - µ ) / σ
1.8808 = ( X - 66 ) / 8
X = 81.05