Speciality Toys

Specialty Toys, Inc. sells a variety of new and innovative childrens 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 chooses an October market entry date.

In order to get toys into its stores by October, Specialty places one-time orders with its manufacturers in June or July of each year. Demand for childrens 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 Teddys hand, the bear begins to talk. A built-in barometer selects 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. Dont 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 Specialtys managers claimed Teddy gave predictions of the weather that were as good as those of many 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 asks you for an analysis of the stock-out probabilities for various order quantities, an estimate of the profit potential, and help with making 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 inventory for $5 per unit.

After reviewing the sales history of similar products, Specialtys senior sales forecaster predicted an expected demand of 20,000 units with a .95 probability that demand would be between 10,000 units and 30,000 units. The forecaster also knows that the demand follows a normal distribution.

1- The sales forecasters prediction can be used to describe a normal probability distribution which approximates the demand distribution. What is the mean of this distribution?

2-Now, with the information above, calculate the standard deviation and round it to an integer number.

Hint: The text gives you information about a specific range for a normal distribution. Determine the critical values (lower and upper boundaries, x₁ and x₂, of the interval you're interested in) for this range in a standard normal distribution and use the formula to convert from normal to standard normal distribution for the calculation.

3-Management wants to order 24,000 units of the product. Calculate the probability that the company will run out of stock, i.e., will actually sell more than 24,000 units.

Provide your answer as probability rounded to two decimal points, e.g., 14.54%

4-Finally, another member of management wants to know how many units should be ordered such that the company will NOT run out of stock with a 95% probability.

In other words, what is the specific number x₀ of sold products for which we know that P(x <= x₀) = 0.95, i.e., with 95% probability product sales will be less than this number? Round your answer to the next number.