Suppose that in the Mens Olympic Ski Team the chest size mea

Suppose that in the Men\'s Olympic Ski Team, the chest size measurements are normally distributed with a mean of 39.8 inches and a standard deviation of 2.05 inches. What is the probability that of 20 randomly selected members of the team, exactly five have a chest size of at least 40 inches? [Hint: first use normal distribution to figure out the probability and then use the binomial to answer the question.]

Solution

Normal Distribution
Mean ( u ) =39.8
Standard Deviation ( sd )=2.05
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
P(X < 40) = (40-39.8)/2.05
= 0.2/2.05= 0.0976
= P ( Z <0.0976) From Standard Normal Table
= 0.5389                  
P(X > = 40) = (1 - P(X < 40)
= 1 - 0.5389 = 0.4611                  

Probability of a chest size of at least 40 inches is = 0.4611

Now, we use Binomial to calculate out of 20 randomly selected members of the team,
exactly five have atleast 40
Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X = 5 ) = ( 20 5 ) * ( 0.4611^5) * ( 1 - 0.4611 )^15
= 0.0303