Question

A grocery store is trying to predict how many packages of toilet paper rolls will be purchased over the next week. They know from their records that each customer has a 60% chance of buying 0 packages, a 30% chance of buying 1 package, an 8% chance of buying 2, and a 2% chance of buying 25. They expect about 150 customers per day.

(a) What is the expected number of packages sold in one day?

(b) What is the probability that the average number of packages per customer is greater than 1 over the course of a week? Assume the store is open 7 days a week.

(c) Would you need a large sample size in order for the probability histogram of the sample mean to be normal? Explain briefly.

Answer 1

(a)

Expected number of packages sold to a customer, E(X) = 0 * 0.6 * 0 + 0.3 * 1 + 0.08 * 2 + 0.02 * 25 = 0.96

Expected number of packages sold in one day = 150 * 0.96 = 144

(b)

Average number of packages per customer = 0.96

E(X^2) = 0.6 * 0^2 + 0.3 * 1^2 + 0.08 * 2^2 + 0.02 * 25^2 = 13.12

Var(X) = E(X^2) - E(X)^2 = 13.12 - 0.96^2 = 12.1984

Standard deviation of average number of packages per customer = = 1.32

By Central Limit theorem, ~ N(0.96, 1.32)

Probability that the average number of packages per customer is greater than 1 over the course of a week

= P( > 1)

= P[Z > (1 - 0.96)/1.32]

= P[Z > 0.0303]

= 0.4879

(c)

Yes, we need a large sample size (greater than 30)  in order for the probability histogram of the sample mean to be normal.