Question

. In San Francisco, 30% of workers take public transportation daily (USA Today, December 21, 2005). In a sample of 10 workers, Let X represents the number of workers taking public transportation daily. [5+5+5+5+10 = 30 points] (a) What is the distribution of X? Give the name and corresponding parameters. (b) Write down the probability mass function of X. (c) In a sample of 10 workers, what is the expected number of workers taking public transportation daily? (d) In a sample of 10 workers, what is the probability that exactly 3 workers take public transportation daily? (e) In a sample of 10 workers, Compute manually the probability that at least 3 workers take public transportation daily.

Answer 1

a) It is a binomial distribution.

n = 10

p = 0.30

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

c) Expected value = n * p = 10 * 0.30 = 3

d) P(X = 3) = 10C3 * (0.3)^3 * (0.7)^7 = 0.2668

e) P(X > 3) = 1 - P(X < 3)

                   = 1 - (P(X = 0) + P(X = 1) + P(X = 2))

                   = 1 - (10C0 * (0.3)^0 * (0.7)^10 + 10C1 * (0.3)^1 * (0.7)^9 + 10C2 * (0.3)^2 * (0.7)^8)

                   = 1 - 0.3828

                   = 0.6172