Question

Define a useful quantity for our expectation about the amount of a time an

in-control process will remain ostensibly in-control, the average run length (ARL), to be the number of samples that will be observed, on average, before a point falls outside control limits. If p is the probability that any given point falls outside the control limits, then:
ARL = 1/p
(Estimating the true ARL may be useful for detecting an out-of-control process, but not
necessarily doing so quickly.) A process is Gaussian with mean 8 and standard deviation 2. The process is monitored by taking samples of size 4 at regular intervals. The process is declared to be out of control if a point plots outside the 3σ control limits on an X-chart. If the process mean shifts to 9, what is the average number of samples that will be drawn before the shift is detected on an X-chart?

Answer 1

First, we need to calculate the control limits for the X-chart. Since the sample size is 4, the standard deviation of the sample mean is:

σ/√n = 2/√4 = 1

The 3σ control limits for the X-chart are:

Upper Control Limit (UCL) = 8 + 3(1) = 11

Lower Control Limit (LCL) = 8 - 3(1) = 5

Next, we need to find the probability that a point falls outside the control limits when the process mean shifts to 9. This can be calculated using the Gaussian distribution:

P(X < 5 or X > 11) = P(Z < (5-9)/2) + P(Z > (11-9)/2) = 0.00135 + 0.00135 = 0.0027

where Z is the standard normal distribution.

Therefore, the average run length (ARL) is:

ARL = 1/p = 1/0.0027 = 370.37

So on average, it will take 370.37 samples before the process mean shift is detected on an X-chart.