Question

. A factory employs several thousand workers, of whom 30% are Hispanic. If the 15 members of the union executive committee were chosen from the workers at random, the number of Hispanics on the committee would have the binomial distribution with n = 15 and p = 0.3.

(a) What is the probability that exactly 3 members of the committee are Hispanic?

(b) What is the probability that 3 or fewer members of the committee are Hispanic?

(c) What is the mean number of Hispanics on randomly chosen committees of 15 workers?

(d) What is the standard deviation σ of the count X of Hispanic members?

Answer 1

X~ bin ( n , p)

Where n = 15 , p = 0.3

Binomial probability distribution is

P(X) = nCx * p^x * ( 1 - p)^n-x

a)

P(X = 3) = ¹⁵C₃ * 0.3³ * (1 - 0.3)¹²

= 0.1700

b)

P(X <= 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

= ¹⁵C₀ * 0.3⁰ * (1 - 0.3)¹⁵ +¹⁵C₁ * 0.3¹ * (1 - 0.3)¹⁴ +¹⁵C₂ * 0.3² * (1 - 0.3)¹³ +¹⁵C₃ * 0.3³ * (1 - 0.3)¹²

= 0.2969

c)

Mean = n p = 15 * 0.3 = 4.5

d)

Standard deviation = sqrt ( n p ( 1 - p) )

= sqrt ( 15 * 0.3 * ( 1 - 0.3 ) )

= 1.7748