Suppose that an X̅ chart is used to control a normally distributed process and that samples of size n are taken every n hours and plotted on the chart, which has k-sigma limits.

(a) Find a general expression for the expected number of samples and time that is taken until a false signal is generated.

(b) Suppose that the process mean shifts to an out-of-control state, say μ₁ = μ₀ + δσ. Find an expression for the expected number of samples that is taken until a false action is generated.

(c) Evaluate the in-control ARL for k = 3. How does this change if k = 2? What do you think about the use of 2-sigma limits in practice?

(d) Evaluate the out-of-control ARL for a shift of 1 sigma, given that n = 5.