A researcher wishes to conduct a study of the color preferen

A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 30% of this population prefers the color green. If 14 buyers are randomly selected, what is the probability that between 3 and 6 (both inclusive) buyers would prefer green? Round your answer to four decimal places.

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

P( X = 3 ) = ( 14 3 ) * ( 0.3^3) * ( 1 - 0.3 )^11
= 0.1943
P( X = 4 ) = ( 14 4 ) * ( 0.3^4) * ( 1 - 0.3 )^10
= 0.229
P( X = 5 ) = ( 14 5 ) * ( 0.3^5) * ( 1 - 0.3 )^9
= 0.1963
P( X = 6 ) = ( 14 6 ) * ( 0.3^6) * ( 1 - 0.3 )^8
= 0.1262
                  
P(3<=X<=4) = P( X = 3 ) + P( X = 4 ) + P( X = 5 ) + P( X = 6 ) = 0.1943+0.229+0.1963+0.1262 =0.7458