### Describe a normal probability distribution

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Case-

Specialty Toys Inc., sells a variety of new and innovative children's toys and believes that the preholiday season is the best to introduce a new toy. Many families us this time to look for new ideas for December holiday gifts. When Specialty has a new toy with good market potential. It chooses an October market entry date.

In order to get toys in its stores by October, Specialty places one-time orders with its manufactures in June or July of each year. Demand for children's toys can be highly volatile. If a new toy catches on, a sense of shortage in the marketplace often increases the demand to very high levels and large profits can be realized. On the other hand, new toys can also flop, leaving Specialty stuck with high levels of inventory that must be sold at reduced prices. The most important question the company faces is deciding how many units of a new toy should be purchased to meet expected sales demand. If too few are purchased, sales will be lost; if too many are purchased, profits will be reduced because of low prices realized in clearance sales.

For the coming season, Specialty plans to introduce a new product called Weather Teddy. This variation of a talking teddy bear is made by a company in Taiwan. When a child presses Teddy's hand, the bear begins to talk. With the aid of a built-in barometer, Teddy says one of five responses that predict the weather conditions. The responses range from "it looks to be a very nice day! Have fun" to "I think it may rain today. Don't forget your umbrella." Tests with the product show that even though it is not a perfect weather perfect predictor, its predictions are surprisingly good. Several of Specialty's managers claimed Teddy gave predictions of the weather that were as good as the local television weather forecasters.

Specialty faces the decision of how many Weather Teddy units to order for the coming holiday season. Members of the management team recommended order quantities of 15,000, 18,000, 24,000, and 28,000. Considerable disagreement concerning the market potential is evidenced by the different order quantities suggested. The product management team has asked for an analysis of the stock-out possibilities for various order quantities an estimate of the profit potential, and to help make an order quantity recommendation. Specialty expects to sell Weather Teddy for \$24, and the cost is \$16 per unit. If inventory remains after the holiday season, Specialty will sell all surplus inventory for \$5 per unit. After reviewing the sales history of similar products, Specialty's senior sales forecaster predicted an expected demand of 20,000 units with a 0.95 probability that demand would be between 10,000 units and 30,000 units.

1. Use the sales forecaster's predictions to describe a normal probability distribution that can be used to approximate the demand distribution. Sketch the distribution and show its means and standard deviation.

2. Compute the probability of a stock-out for the order quantities suggested by members of the management team.

3. Compute the projected profit for the order quantities suggested by the management team under three scenarios: worst case in which sales equal 10,000 units, most likely case in which sales equal 20,000 units, and best case in which sales equal 30,000 units.

4. One of Specialty's managers felt that the profit potential was so great that the order quantity should have a 70% chance of meeting demand and only a 30% chance of any stock-outs. What quantity would be order under this policy, and what is the projected profit under the three sales scenarios?

5. Provide your own recommendations for an order quantity and note the associated profit projections. Provide a rationale for your recommendation.

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