In recent survey 80 of the community favored building a poli

In recent survey, 80% of the community favored building a police substation in their neighborhood. If 20 citizens are chosen, what is the mean and standard deviation for number favoring the substation? Also, find the probablity that 12 or fewer people favor building a police substation in thier neighborhood.

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

Mean ( np ) =16
Standard Deviation ( npq )= 20*0.8*0.2 = 1.7889

P( X < = 12) = P(X=12) + P(X=11) + P(X=10) + .... + P(X=2) +P(X=1) + P(X=0)
= ( 20 12 ) * 0.8^12 * ( 1- 0.8 ) ^8 + ( 20 11 ) * 0.8^11 * ( 1- 0.8 ) ^9 + ( 20 10 ) * 0.8^10 * ( 1- 0.8 ) ^10 + ....... + ( 20 3 ) * 0.8^3 * ( 1- 0.8 ) ^17 + ( 20 2 ) * 0.8^2 * ( 1- 0.8 ) ^18 + ( 20 1 ) * 0.8^1* ( 1- 0.8 ) ^19+
( 20 0 ) * 0.8^0 * ( 1- 0.8 ) ^20
= 0.0321