Consider that the standard practice is for the quality control lab to select a sample of ns ball bearings, compute the speci?c value for the sample mean, x¯, and plot it on a chart with the following characteristics:

a center line representing μ = 10.00; an upper limit line set at (10 + 3σ/√ns) and
a lower limit set at (10 - 3σ/√ns). The process is deemed to be performing "as expected" if the value obtained for x¯ falls within the limits.

(i) For a sample of 4 ball bearings, where are the upper and lower limits lines located? What is the probability of x¯ falling outside these limits when the process is in fact operating "as expected".

(ii) If a process "disturbance" shifted the true mean diameter for the manufactured ball bearings to 10.10 mm, what is the probability of detecting this shift when the result obtained from the next sample of 4 ball bearings is analyzed?

State any assumptions needed to answer these questions.