Oxford Cereals fills thousands of boxes of cereal during an eight-hour shift. As the plant operations manage, you are responsible for monitoring the amount of cereal placed in each box. To be consistent with package labeling, boxes should contain a mean of 368 grams of cereal. Because of the speed of the process, the cereal weight varies from box to box, causing some boxes to be under-filled and others overfilled. If the process is not working properly, the mean weight in the boxes could vary too much from the label weight of 368 grams to be acceptable. The acceptable standard deviation of this process is 15 grams. Because weighing every single box is too time-consuming, costly, and inefficient, you must take a sample of boxes. For each sample you select, you plan to weight the individual boxes and calculate a sample mean. You need to determine the probability that such a sample mean could have been randomly selected from a population whose mean is 368 grams. Based on your analysis, you will have to decide whether to maintain, alter, or shut down the cereal-filling process.

a. Suppose that a sample of 25 boxes has been taken, and you find the mean to be 362.3 grams. Does the filling process appear to be working correctly? Assume you are 95% confident in your statistical process.

b. Suppose that you have never tested this process, so do not know the standard deviation of the filling process. A sample of 40 boxes has been taken, and you find the mean to be 362.3 grams and a standard deviation of 2.3 grams. Does the filling process appear to be working correctly? Assume you are required to be 95% confident in your statistical process.