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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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