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# Assignment 4 - 2012

### From Statistics for Engineering

Due date(s):
| 06 February 2012, noon |

(PDF) | Assignment questions |

(PDF) | Assignment solutions |

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# Question 1 [1]

Describe what \(\overline{S}\) and \(a_n\) represent in the derivation of the Shewhart chart control limits.

# Question 2 [4; 6 (for 600-level)]

Plant data from a flotation cell are provided on the data set website. One would normally calculate a monitoring chart's limits from a much longer period of data than provided in this data set. We will however use this small data set to illustrate the principle.

Feel free to use *any* software to answer these questions. There is no need to include your code with the solution; just clear explanations of what you are doing, and the corresponding plots.

- Plot the time-series data for the entire sequence of observations for the
`Feed rate`column, which represents the tons of ore fed to the flotation circuit per hour. What do you observe in these data? - Use all data on 15 December 2004 (points 1 to 479) from the
`Feed rate`variable as your phase 1 data. You have no other process information to go on, so make any assumptions as required. Iteratively prune any outliers to settle on a reasonable set of monitoring parameters. A subgroup size of 4, representing 2 minutes of operation, should be used. - Use data from 16 December 2004 (points 480 onwards) from the
`Feed rate`variable as your phase 2 (i.e. testing) data. Show the performance of your monitoring parameters calculated in part 1 on these phase 2 data. - Explain why the monitoring system works (or doesn't work) as you expect.
**600-level students**(extra-credit for 400-level students): implement an alternative monitoring chart using these same data. Describe your calculations.

# Question 3 [1]

Your process with Cpk of 2.0 experiences a drift of \(1.2\sigma\) away from the current process operating point towards the closest specification limit. What is the new Cpk value; how many defects per million items did you have before the drift? And after the drift?

# Question 4 [2]

*From the 2010 midterm. Show full calculations please.*

You need to construct a Shewhart chart. You go to your company's database and extract data from 10 periods of time lasting 6 hours each. Each time period is taken approximately 1 month apart so that you get a representative data set that covers roughly 1 year of process operation. You choose these time periods so that you are confident each one was from in control operation. Putting these 10 periods of data together, you get one long vector that now represents your phase 1 data.

- There are 8900 samples of data in this phase 1 data vector.
- You form subgroups: there are 4 samples per subgroup and 2225 subgroups.
- You calculate the mean within each subgroup (i.e. 2225 means). The mean of those 2225 means is 714.
- The standard deviation within each subgroup is calculated; the mean of those 2225 standard deviations is 98.

- Give an unbiased estimate of the process standard deviation?
- Calculate lower and upper control limits for operation at \(\pm 3\) of these standard deviations from target. These are called the action limits.
- Operators like warning limits on their charts, so they don't have to wait until an action limit alarm occurs. Discussions with the operators indicate that lines at 590 and 820 might be good warning limits. What percentage of in control operation will lie inside the proposed warning limit region?