Suppose that a certain industrial process can be either in control or out of control, and that at any specified time the prior probability that it will be in control is 0.9, and the prior probability that it will be out of control is 0.1. A single observation X of the output of the process is to be taken, and it must be decided immediately whether the process is in control or out of control. If the process is in control, then X will have the normal distribution with mean 50 and variance 1.If the process is out of control, then X will have the normal distribution with mean 52 and variance 1.
If it is decided that the process is out of control when in fact it is in control, then the loss from unnecessarily stopping the process will be $1000. If it is decided that the process is in control when in fact it is out of control, then the loss from continuing the process will be $18,000. If a correct decision is made, then the loss will be 0. It is desired to find a test procedure for which the expected loss will be a minimum. For what values ofX should it be decided that the process is out of control?