In a university elevator there is a sign indeicating a 16per

In a university elevator there is a sign indeicating a 16-person limit as well as a weight limit of 2500 lbs. suppose that the weight of students, faculty, and staff is approximately Normally distributed with a mean weight of 150 lbs and a standard deviation of 27 lbs.

a. what is the probablilty that the random sample of 16 people in the elevator will exceed the weight limit?

b. suppose we selected a SRS of 12 groups (16 people in each) what is the probability, at most 25% of the groups will exceed the maximum weight limit.

Solution

Normal Distribution
Mean ( u ) =150
Standard Deviation ( sd )=27
Normal Distribution = Z= X- u / sd ~ N(0,1)              


a)
P(X > 156.25) = (156.25-150)/27/ Sqrt ( 16 )
= 6.25/6.75= 0.9259
= P ( Z >0.9259) From Standard Normal Table
= 0.1772                  
b)
P( Exceeding the weight limit is ) = 0.1772

P( atleast 25% are exceed weight limit) = P( X > = 4 )

P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 12 3 ) * 0.1772^3 * ( 1- 0.1772 ) ^9 + ( 12 2 ) * 0.1772^2 * ( 1- 0.1772 ) ^10 + ( 12 1 ) * 0.1772^1 * ( 1- 0.1772 ) ^11 + ( 12 0 ) * 0.1772^0 * ( 1- 0.1772 ) ^12
= 0.8514
P( X > = 4 ) = 1 - P( X < 4) = 0.1486