Suppose that the proportion of statistics students who are s

Suppose that the proportion of statistics students who are sophomores is 20%. We take a random sample of 10 statistics students.

(1) Use the binomial formula to find the probability that the sample contains exactly 2 sophomores.

(2) Use the binomial  probability Table B to find the find the probability that the sample contains at least 3 sophomores.

(3) What is the most likely number of sophomores in the sample? Explain.

(4) Find the mean, variance and standard deviation of the number of sophomores. Interpret the mean.

Solution

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

a)
P( X = 2 ) = ( 10 2 ) * ( 0.2^2) * ( 1 - 0.2 )^8
= 0.302

b)

P( X < 3) = P(X=2) + P(X=1) + P(X=0)   
= ( 10 2 ) * 0.2^2 * ( 1- 0.2 ) ^8 + ( 10 1 ) * 0.2^1 * ( 1- 0.2 ) ^9 + ( 10 0 ) * 0.2^0 * ( 1- 0.2 ) ^10
= 0.6778
P( X > = 3 ) = 1 - P( X < 3) = 0.3222

c)
P ( Z < x ) = 0.99
Value of z to the cumulative probability of 0.99 from normal table is 2.326
P( x-u/s.d < x - 2/1.2649 ) = 0.99
That is, ( x - 2/1.2649 ) = 2.33
--> x = 2.33 * 1.2649 + 2 = 4.9422                  
It is 5

d)
Mean ( np ) =10 * 0.2 = 2
Standard Deviation ( npq )= 10*0.2*0.8 = 1.2649