Use the sales forecaster's prediction to describe a normal probability distribution

Statistic, Assignment 2

Specialty Toys, Inc. sells a variety of new and innovative children’s toys. Management learned that the preholiday season is the best time to introduce a new toy, because many families use this time to look for new ideas for December holiday gifts. When Specialty discovers a new toy with good market potential, it choses an October market entry date.

In order to get toys in its stores by October, Specialty places one-time orders with its manufactures in 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 high levels and large profits can be realized. However, 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 anticipated 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. A built-in barometer selects one of five responses that predict 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 predictor, its predictions are surprisingly good. Several of Specialty’s managers claimed Teddy gave predictions of the weather as good as local television weather forecasters.

As with other products, Specialty faces the decision of how many Weather Teddy units to order for the coming holiday season. Members of the management team suggested order quantities of 15,000, 18,000, 24,000 or 28,000 units. The wide range of order quantities suggested indicates considerable disagreement concerning the market potential. The product management team asked you for an analysis of the stock-out probabilities 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 based on a cost of $16 per unit. If inventory remains after the holiday season, Specialty will sell all surplus inventories for $5 per unit. Specialty’s sales forecaster predicted the demand for Weather Teddy with an expected value of 20,000 units and a standard deviation of 5,100 units.

Managerial Report

Prepare a managerial report that addresses the following issues.

1. Use the sales forecaster’s prediction to describe a normal probability distribution that can be used to approximate the demand distribution. Find the probability that the demand will vary between two most extreme order quantities suggested by members of the management team.

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

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

4. One of Specialty’s managers felt that the profit potential was so great that the order quantity should have a 60% chance of meeting demand and only a 40% chance of any stock-out. What quantity would be ordered under this policy, and what are the projected profits under the three sales scenarios?

5. Provide your own recommendation for an order quantity and list the associated profit projections under the three sales scenarios. Provide a rationale for your recommendation.

Instructions

1. Re: Issue 1. Let D be the demand, and assume that D is normally distributed with mean µ = 20,000, and standard deviation σ = 5,100. Compute P(15,000 ≤ D ≤ 28,000). To get a full credit, use Excel function NORM.DIST to find P(D ≤ x) for a specified value of x; see pages 285-286. You will get a partial credit if you apply the z table.

2. Re: Issue 2. For example, the probability of a stock-out for the suggested order quantity 15,000 is P(D > 15,000) = 1 – P(D ≤ 15,000). Again, use NORM.DIST to find P(D ≤ x).

3. Re: Issue 3. For example, assuming the order quantity of 15,000 and the worst case scenario, the projected profit is 10,000($24) + 5,000($5) – 15,000($16) = $25,000. On the other hand, assuming the order quantity of 18,000 and the best case scenario, the projected profit is 18,000($24) – 18,000($16) = $144,000. You are expected to show all 4(3) = 12 projected profits.

4. Re: Issue 4. Let Q be the 60% - 40% order quantity. Using the relation P(D Q) = 0.6, find Q. To get a full credit, use Excel function NORM.INV; see page 286. You will get a partial credit if you apply the z table. After finding Q, compute the corresponding projected profits under the three demand scenarios.

5. Re: Issue 5. If Q* is your recommended order quantity, compute the corresponding projected profits under the three demand scenarios.

Your managerial report should be written in MS Word and should address all 5 issues. (Do not attach any Excel files; instead you may paste Excel results.) Of course, the important element of this report should be your own recommendation (Issue 5) with the provided rationale.

Attach your managerial report in Blackboard, and bring its hard copy to the class on the designated day,