Question

The average number of ice cream cones sold at the local ice

cream stand is 1000 per month. The standard deviation is 200. The
historical demand distribution has a nice bell shape, and so it
seems reasonable to model the demand as if it were normally
distributed.
Suppose the stand has an inventory of 1100 cones available to
meet the demand in one month. What is the probability that the
stand will run out of cones?
How many cones should the stand carry in order that its stockout
probability in one month is no more than 0.05?
Suppose demand in each month is independent from demand in any
other month. What is the probability that demand over a three month
period exceeds 3500 ice cream cones?

Answer 1

To determine the probability of running out of cones, calculate the z-score and use a standard normal distribution. Carry at least 670 cones to have a stockout probability no more than 0.05. Use the Central Limit Theorem to calculate the probability of demand exceeding 3500 cones over a three-month period.

Step-by-step explanation:

To determine the probability that the ice cream stand will run out of cones, we need to calculate the z-score. We can use the formula: z = (X - µ) / σ, where X represents the inventory available, µ represents the average demand, and σ represents the standard deviation. Plugging in the values, we get z = (1100 - 1000) / 200 = 0.5. We can then use a standard normal distribution table or a calculator to find the probability associated with a z-score of 0.5, which is approximately 0.6915.

To find the number of cones the stand should carry to have a stockout probability no more than 0.05, we need to find the z-score associated with a cumulative probability of 0.05. Using a standard normal distribution table or a calculator, we find that the z-score is approximately -1.645. We can then use the formula z = (X - µ) / σ and solve for X, plugging in the values, to find X = -1.645 * 200 + 1000 = 669. We round up to the nearest whole number, so the stand should carry at least 670 cones.

To calculate the probability that demand over a three-month period exceeds 3500 ice cream cones, we need to model the distribution of the sum of demands over the three months using the Central Limit Theorem. The average demand over a three-month period is 1000 * 3 = 3000 cones, and the standard deviation of the sum is √(200^2 × 3) = 346.41 cones. We can then use the z-score formula again to find the z-score for a demand of 3500 cones, z = (3500 - 3000) / 346.41 = 1.558. Using a standard normal distribution table or a calculator, we find the probability associated with a z-score of 1.558, which is approximately 0.9394. Therefore, the probability that demand over a three-month period exceeds 3500 ice cream cones is approximately 0.9394.