Exercise on Process Variation

This exercise is meant to illustrate by simple calculations that tampering a stable process (i.e., intervening in process that is statistically in control) increases process variation. It is a one-dimensional simplification of the Deming funnel experiment.

The process consists of throwing one arrow at a time with the Dartec (see picture above). The height of the kth arrow before throwing is denoted by Tk. After a throw we are allowed to change the height of the Dartec. The height of the kth arrow on the dart board is denoted by Pk. The goal is to minimise the variation of the height Pk around a target value T.

Assume that the values of Pk ’s are a sample from an unknown distribution with mean Tk and standard deviation s, i.e. Pk =Tk + ek where the ek’s are independent, have mean zero and have a common standard deviation s.

We consider three strategies:
– Strategy 1: do not touch the Dartec, i.e. Tk = T1=T .
– Strategy 2: adjust the height of the Dartec by adding the deviation from the target of the previous throw, i.e. Tk = Tk-1 + (T - Pk-1). An alternative description of this strategy is to place the Dartec on the same height as the target and then adjust for the previous difference between height of the Dartec and height after the throw.
– Strategy 3: put Dartec on the same height as the target and then adjust for the previous deviation of the target, i.e. Pk = 2T - Pk-1.

a)  What are the mean and variance of Pk under strategy 1?

b)  What are the mean and variance of Pk under strategy 2?

c)  What are the mean and variance of Pk under strategy 3?

d)  Compare the results of a), b) and d). What is your conclusion?

e)  Open the Excel sheet below. Generate in Excel or Statgraphics a random sample of size 100 from a normal distribution with mean 0 and variance 1.2 . Copy this data set and paste it into the Excel sheet below. Relate the graphics with your results in e). Note: if you cannot do statistical analyses in Excel, then go to Tools -> Add-ins and add the Analysis Toolpak (see PowerPoint file below with detailed instructions).



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