Question

3. Suppose that to encourage alumni giving. Holy Cross hires students to call alumni to ask them to make a gift to the college. Typically, about 20% of students who are hired feel uncomfortable asking strangers for money and quit while 80% of students who are hired enjoy the opportunity to connect with alumni and stay in the position through the semester. Suppose that this semester the college hired six students to call alumni. (a) What is the probability that at least two of the six will quit? (b) What is the probability that at least two of the six will stay in the position through the semester? (c) What is the expected number of students who will quit? (d) What is the standard deviation of the number of students who will quit?

Answer 1

a) p = 0.2

n = 6

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

              = 1 - (6C0 * 0.2⁰ * 0.8⁶ + 6C1 * 0.2¹ * 0.8⁵)

              = 1 - 0.6554

              = 0.3446

b) p = 0.8

n = 6

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

              = 1 - (6C0 * 0.8⁰ * 0.2⁶ + 6C1 * 0.8¹ * 0.2⁵)

              = 1 - 0.0016

              = 0.9984

c) n = 6

p = 0.2

Expected value = n * p = 6 * 0.2 = 1.2

d) standard deviation = sqrt(n * p * (1 - p)) = sqrt(6 * 0.2 * 0.8) = 0.98