A large toy-store chain is considering the launch of a new toy, currently codenamed Cowboy23, in time for Christmas 2023.

The question the company faces is deciding how many units to order from its manufacturers to meet anticipated sales demand. If too few are ordered, sales will be lost (a stock-out). If too many are ordered, losses will be incurred on because of low prices realised in post-Christmas clearance sales. The companys sales analyst predicts an expected demand of 25,000 units with a 0.95 probability that demand will be between 15,000 units and 35,000 units. The four members of the management team have suggested order quantities for Cowboy23 of 20,000, 24,000, 27,000 and 32,000 units. The manufacturing cost is 20 per unit.

The company expects to sell Cowboy23 before Christmas for 30 per unit. After the Christmas period, the company will sell surplus inventory for 10 per unit.

Prepare a report that addresses the following:

1. Using the sales analysts predictions, calculate the mean and standard deviation of a normal distribution that will approximate the probability distribution for demand.

2. Calculate the probability of a stock-out for the order quantity suggested by each member of the management team.

3. Calculate the projected profit for the order quantities suggested by the members of the management team under three scenarios: worst case, sales = 15,000 units; most likely case, sales = 25,000 units; best case, sales = 35,000 units.

4. Suppose the payment to the manufacturer is made at the end of July 2023. Assume half the Christmas sales revenue is received at the end of November, half at the end of December, and the after-Christmas sales revenue is received at the end of January 2024. Assume 32,000 units are ordered, and Christmas sales match the sales analysts expected demand figure. What is the projected profit at July 2023 prices if a discount rate of 0.25% per month is applied to the revenues?