# Business: Decision Justifications Using Z-score And Statistics

Assume that you own a small factory. A critical piece of machinery in your factory will need to be replaced in 180 days. If the machinery does not show up on time, you will need to shut down until it arrives. This might cause you to permanently lose customers.When you order the part you will need to pay the $500,000 in advance. That is a lot of money for your small business.

If you keep the money in the bank, it will earn interest each month. If you spend the money now, it will leave you with very little money on hand, and you might have to borrow money to make payroll. You know from past experience that the delivery times are normally distributed with a mean of 45 days and a standard deviation of 15 days. When should you order the part? It is your company, but you must write up an explanation for your actions that convinces your investors that your actions are best. (Unfortunately, the investors cannot afford $500,000 at this time.)

NOTE: There are many right answers to this question. Grading will be based on your explanations and your defense of your choice. .

Be sure to include

1) Explain why 135 days from now is the date to order it if you want to be 50% sure of getting the delivery on time.

2) Explain to the investors why you cannot be 100% sure that the part arrives in time.

3) How sure do you want to be that the machinery arrives on time? Explain your answer clearly, and explain why you chose that figure. Remember, you need to convince your investors.

(the answer here cannot be 50%)

4) Based on how sure you want to be, calculate when you need to order the part. Explain all calculations clearly, because your investors need to understand what you did. They are bright and understand math, but you will have to explain the statistics to them. This needs to include z-scores and a calculation of x based on z.

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#### Solution Preview

The delivery times are normally distributed so we can use the properties of a normal distribution. Also, the delivery time distribution has a mean of 45 days and a standard deviation of 15 days.

Let's start with the first part of the question:

1. Because the delivery time has a normal distribution, 50% of the delivery times lie below the mean, 50% above the mean. The mean of this normal distribution is 45. That is, if we order 45 days ahead of the due date, we can be 50% sure that the delivery will be on time. 45 days ahead of the due date means 180-45=135 days from now.

Now, let's think about the 120th day.

120 days from not means 180-120=60 days before the due date. To find the delivery times that lie below 60 on ...

#### Solution Summary

This solution provides a detailed, step by step calculation of the given statistics problem.