Suppose that the proportion of statistics students who are sophomores is 20%. We

Question

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.

Explanation / Answer

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

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