Chapter 16.1, Problem 5E

If a process variable is normally distributed, in the long run virtually all observed values should be between µ − 3σ and µ + 3σ, giving a process spread of 6σ.

a. With LSL and USL denoting the lower and upper specification limits, one commonly used process capability index is $C_p=\left(USL–LSL\right)/6\sigma$ The value C_p = 1 indicates a process that is only marginally capable of meeting specifications. Ideally, Cp should exceed 1.33 (a “very good” process). Calculate the value of Cp for each of the cork production processes described in the previous exercise, and comment.

b. The C_p index described in (a) does not take into account process location. A capability measure that does involve the process mean is

$C_{pk}=\text{min}\left\{(\text{USL-}\mu \text{)}/3\sigma ,(\mu \text{-LSL})/3\sigma \right\}$

Calculate the value of C_pk for each of the corkproduction processes described in the previous exercise, and comment. [Note: In practice, m and s have to be estimated from process data; we show how to do this in Section 16.2]

c. How do C_p and C_pk compare, and when are they equal?