Statistics 101: Homework 6

Here’s an answer to the Chapter 2 case that we liked. It works well because:

- It started the key idea, high variability on the right line after week 10 immediately.
- It responded to the manager’s question about equal averages clearly and promptly.
- It suggested causes without jumping to unproven conclusions.
- It separated the essentials that the manager’s need to know from the more technical justification.
- It wasted very few words.

Executive Summary

By analyzing the data you provided on steak weights from your right and left production lines, we found that after week 10, when production was near full capacity, the weights of steaks cut on the right line varied enormously. Therefore, even though the average weight of steak cut on the right line were well within your acceptable range, a great number of the steaks produced weighted either well above or well below the target 12 oz. Since just as many of the steaks weighed too much as too little, the average turned out to be on target. Steaks cut on the left line during the entire 25 weeks and those cut on the right during the first 10 week varied very little in weight and the average amount of variation stayed the same over the period.

We believe that this increase in production unpredictability on the right line was cased by the increase in production speed after the tenth week. Without knowing more about your production process we can only speculate as to the reasons why this might be so. Perhaps a piece of equipment on the right line, such as scale, no longer functions well at high speeds, or perhaps the worker on the right line have not been as well trained.

Statistical Justification

The x-bar charts attached (which plot the average weight of steaks produced each week clearly show that steaks from both production lines weighed just over 12 oz. on the average, as your letter indicated. A time sequence plot of the left line data (which plots the exact weight of each of the 250 steaks sampled shows that steaks produced on the left line varied consistently in the acceptable range of 11.6-12.6 ounces. However, a similar time sequence plot of the right line data shows a noticeable lump after the tenth week, an indication that the weight of steaks had begun to vary much more widely. A comparison of S charts (which plot standard deviation over time) of the left and right line data confirms this theory. While standard deviations for the left line remains small and stable at approximately 0.2, standard deviation for the right line jumps from a respectable 0.15 in the tenth week to a whopping 0.67 in the fourteenth week. Although it continues to fluctuate, the average standard deviation is much higher after week 10.

Notes on the "MisSteak" Case, Chapter 2

The essential idea is difference in variation between the two lines. One good approach would be to construct separate R or S charts for the two lines. The Left line has a stable pattern – no evident trend – and relatively low variation. The Right line shows a jump in variability at about week 10. A control chart is an excellent way to show the jump. Apparently, once the production process approached full capacity, the Right line got somewhat out of control, and variability increased considerably.

Simply listing ranges or standard deviations is also a workable approach. The increase in variability for the Right line should once again be apparent if weekly ranges and standard deviations for the two lines are compared.

Looking at averages is __not__ as good a way to get at the essential problem. Granted, the Right line means bounce around more than the left line means, as a consequence of the greater variability of Right line measurements. However, studying means is, at best, an indirect way to understand variability.

A subtler point has to do with the effect of greater Right line variability. This variation will translate to greater variation in the total weight of 24-steak packs. Presumably, there will be overweight packs as well as underweight packs. However, it is very unlikely that customers will complain about receiving too much for their money!

There does not seem to be evident skewness or an excess of outliers to explain the problem. Week to week boxplots, for example, do not seem unreasonably shaped.

Note that the rough scale did not work perfectly. A few of the individual weights fall outside the nominal range.