Question

Lee’s bookstore is preparing for homecoming by ordering “Lee Homecoming” t-shirts with the year on them. With homecoming in November, the bookstore must place a one-time t-shirt order in September. The most important question is how many t-shirts to purchase. If too few are purchased, profits will be lost. If too many are purchased profits will be reduced because of low prices realized in clearance sales.

For the coming year, management has some disagreement on order quantities. Various members are suggesting quantities of 3,000, 3,600, 4,800, and 5,600 shirts. The wide range suggests some disagreement. They have asked you for an analysis of the stock-out probabilities for the various order quantities, an estimate of the profit potential, and help with making some an order quantity recommendation.

Shirts cost $3.2 and they expect them to sell at $5 during homecoming. If the shirts cannot be sold during homecoming, they will have to be sold on clearance for $1. The bookstore wants to maximize profits.

Forecasts expect demand for the shirts to be around 4,000 with a 0.95 probability that demand will be between 2,000 and 6,000 shirts.

1.     Describe the normal probability distribution that can be used to approximate the demand distribution. Create a graph of the distribution showing the mean and standard deviation.

Answer 1

The probability density function of the normal distribution with mean and standard deviation , normal distribution is defined as

To approximate the demand distribution with normal density we need the mean and standard deviation . But we have given that, forecasts expect demand for the shirts to be around 4,000 with a 0.95 probability that demand will be between 2,000 and 6,000 shirts.

Thus since mean gives the central tendency of the distribution and it is given that, expect demand for the shirts to be around 4,000,

Also 0.95 probability that demand will be between 2,000 and 6,000 shirts. But  95% of the area of a normal distribution is within 1.96 standard deviations of the mean,

Thus we can approximate demand using

The plot of the distribution