A small grocery store sells fresh produce, which it obtains from a local farmer. During the strawberry season, demand for fresh strawberries can be reasonably approximated using normal distribution with a mean of 40 quarts per day and a standard deviation of 6 quarts per day. Excess costs run 35 cents per quart. The grocery orders 49 quarts per day.
a)What is the implied cost of shortage per quart?
b)Why might this be reasonable figure?
To find the implied cost of shortage per quart, we need to calculate the probability of running out of strawberries and then multiply it by the excess cost per quart.
a) To calculate the probability of running out of strawberries, we need to find the probability of demand exceeding the quantity ordered. Since we know that the demand follows a normal distribution with a mean of 40 quarts per day and a standard deviation of 6 quarts per day, we can use the z-score formula to standardize the values.
The formula for calculating the z-score is:
z = (X - μ) / σ
Where:
X = the quantity ordered (49 quarts per day)
μ = the mean (40 quarts per day)
σ = the standard deviation (6 quarts per day)
Substituting the values, we get:
z = (49 - 40) / 6 = 1.5
Now we need to find the area under the normal distribution curve to the right of the z-score of 1.5. We can use a standard normal distribution table or an online calculator to find this area.
Using a standard normal distribution table, we find that the probability (P) is approximately 0.0668. This means there is a 6.68% chance that the demand will exceed the quantity ordered.
Finally, we can calculate the implied cost of shortage per quart by multiplying the excess cost per quart (35 cents) by the probability of shortage:
Implied cost of shortage per quart = excess cost per quart * probability of shortage
= 0.35 * 0.0668
≈ 0.0233 cents per quart
b) This figure might be reasonable because it captures the cost associated with shortage. If the grocery store runs out of strawberries, it would not be able to meet the demand, potentially leading to dissatisfied customers or loss of sales. The cost of shortage takes into account the potential revenue loss and the negative impact on the store's reputation. By factoring in the excess cost per quart and the probability of shortage, the grocery store can estimate the financial impact of not having enough strawberries in stock.