# Six sigma

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For Statistics for Quality Control

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##### Solution Summary

1. A quality characteristic is monitored by a control chart designed so the probability that a certain out-of-control condition will be detected on the first sample following the shift to that state is 1 - β. Find the following:

a) The probability that the out-of-control condition will be detected on the second sample following the shift.

b) The probability that the out-of-control condition will be detected on the mth sample following the shift.

c) The expected number of sub-groups analyzed before the shift is detected.

d) The probability that the first sample following the shift produces a statistic that plots inside the control limits.

e) The probability that the first sample following the shift produces a statistic that plot inside the control limits.

f) The probability that two consecutive samples produce statistics that plot out-side of the control limits.

g) The probability that at least one of the next m samples yields values of the test statistics that plot out-side the control limits.

##### Solution Preview

We can describe such design using a tree diagram.

<br>Here, we define all events that can happen and assign a probability of an individual event for each branch.

<br>For example, in the first layer (or sample1), there are two probable events:

<br>- an out-of-control condition is detected, probability=1 - β

<br>- an out-of-control condition is not detected, probability=β

<br>Notice that ...

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