A cereal manufacturer has a machine that fills the boxes. Boxes are labeled “16 oz,” so the company wants to have that much cereal in each box. But since no packaging process is perfect, there will be minor variations. If the machine is set at exactly 16 oz and the

Normal distribution applies (or at least the distribution is roughly symmetric), then about half of the boxes will be underweight, making consumers unhappy and exposing the company to bad publicity and possible lawsuits. To prevent underweight boxes, the manufacturer has to set the mean a little higher than 16.0 oz. Based on their experience with the packaging machine; the company believes that the amount of cereal in the boxes fits a Normal distribution with a standard deviation of 0.2 oz. The manufacturer decides to set the machine to put an average of 16.3 oz in each box.

**(a) **What fraction of the boxes will be underweight? **(Total: 10 Marks)**

**Steps**

- State the variable and the objective.
**(2)** - Explain whether a Normal distribution is appropriate or not. If appropriate, state the distribution.
**(3)** - By using an appropriate distribution, and corresponding diagrams to show your working, find the proportion of boxes that will be underweight.
**(5)**

The Company’s lawyers insist that the current proportion of underweight boxes is unacceptable and it may create problems. They insist that no more than 4% of the boxes can be underweight. So the company needs to set the machine to put a little more cereal in each box.

**(b)** What mean setting do they need? **(Total: 10 Marks)**

**Steps**

- State the variable and the objective.
**(2)** - Determine the appropriate value of
*z*-score that will leave no more than 4% of the boxes weighing less than 16 oz.**(3)** - Using the
*z*-score found in part**(b)**ii above, determine the mean weight that will enable the company to keep no more than 4% of the boxes weighing less than 16 oz.**(5)**