Page 221 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

August 31, 2006
2:58
Char Count= 0
JWBK119-13
206 A Graphical Approach to Obtaining Confidence Limits of C pk
13.6 ILLUSTRATIVE EXAMPLES
We use the following examples to demonstrate the usage of AM confidence limits for
C pk under two different scenarios.

13.6.1 Example 1
A company has a well-established process for supplying a certain product for its
customer. To ensure low incoming defects, the customer specifies a minimum C pk -
value of 1.8, with a lower confidence limit of 1.5 to cater for sampling variability. A
process capability study conducted on the process reveals the following information:
n = 100,USL = 30,LSL = 12, x = 1.27ands = 1.5.Assumethatthedesiredconfidence
level for all calculations is 95 % (α = 0.05).
Substituting these data in equations (13.1)--(13.4) will give

30 − 12
ˆ
C p = = 2.0,
6 × 1.5
1
m = (30 + 12) = 21,
2
ˆ |21 − 21.27|
k = = 0.03,
(30 − 12)/2
ˆ
C pk = (1 − 0.03) 2.0 = 1.94.
ˆ
Since k < 0.1, Table 13.5 recommends C pk1 as the lower AM confidence limit for C pk .
The k-value is reasonable in this case for a well-established and optimized process.
From equation (13.6), C is
p
χ 2
100 − 1,0.05/2
C = √ × 2.0 = 1.684.
p
100 − 1
From equation (13.14), C pk1 is

C = (1 − 0.03) 1.684 = 1.633.
pk1
The company can thus safely claim to the customer that this process meets the mini-
mum C pk value and lower confidence limit value.

13.6.2 Example 2
Another process is a new one acquired by a supplier. It is desired to qualify the
process by running a test production. The customer specifies a minimum C pk value of
1.3, with a lower confidence limit of 1.0. Since process is new, it is yet to be optimized
and controlled for production. A process capability study conducted for the process
records the following data: n = 50, USL = 20.8, LSL = 10, x = 17.2 and s = 1.2.