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-beta. Find the following:
a) The probability that the out-of-control condition will be detected on the n'th sample following the shift.
b) The expected number of subgroups analyzed until the shift is detected.
c) The probability that the first sample following the shift produces a statistics that plots inside the control limits.
d) The probability that n consecutive samples produce statistics that plots outside of the control limits.
e) The probability of detecting the shift after the nth sample following the shift, which means the shift is detected in the (n+1)th, or (n+2)th, ... sample
f) The probability that at least 1 out of n consecutive samples produce statistics that plots outside of the control limits.