Cola Company has recently installed new bottling equipment. As the head of quality control, management would like a report on the new equipment. In particular, management wants to be sure that the new bottling machinery is filling the bottles precisely. Based on sampling from the previous equipment, the typical standard deviation in the bottling process was 0.18. The current machine should bottle cola in 16 oz bottles, which is what the rm advertises when they sell the product. If the bottling machinery is working well, management will consider expanding the bottling equipment to other factories across the country. If the machinery is not working well, management will need to stop production until the machinery can be corrected. Attached are random samples taken from the first week of operation for this new machine. Management has asked for a full report on the new machinery. The report you should submit to management should include at a minimum the following:

1. Provide descriptive statistics of the sample data. Make sure management has a clear understanding of the key characteristics of the data, including location, variation, and shape.

2. Discuss your assumptions regarding the treatment of the historical standard deviation, and how it relates to the sample standard deviation. Is it reasonable to use the historical standard deviation for the new bottling equipment?

3. Formulate and develop an appropriate statistical hypothesis consistent with evaluating the performance of the new machinery. Make sure to include a thorough discussion of type I and type II errors, as well as risks associated with these types of errors. How do you intend to balance these risks? Justify your approach, including selecting an appropriate significance level.

4. Conduct a hypothesis test for each sample included in the data. Discuss and interpret the results.

5. Develop quality control limits. That is, to further quality control going forward, compute the limits for the sample mean around the desired population parameter so that as long as each sample mean is within those limits, the process will be considered to be operating satisfactorily. Or put differently, develop bounds such that if the sample mean is above or below the upper or lower limits, corrective action will be undertaken.

6. Discuss your approach and reasoning for developing your particular quality control limits, including any caveats. For example, how would a change in the significance level affect the quality control structure you are recommending? Discuss the costs and benefits of changing the level of significance.

7. Provide a recommendation to management about whether the machinery should be expanded to additional factories.

**Subject Mathematics Advanced Statistics**