Confidence interval and tests on process
capability ratios
• Note that Cˆp or Cˆpk is a simply point estimate. It is more necessary to report the
Home Page
Title Page
JJ
II
J
I
Page 1 of 37
confidence interval for PCR.
Go Back
Full Screen
Close
Quit
• For the “first generation” ratio Cp, replace σ by S, Cˆp = (U SL − LSL)/(6S),
the 100(1 − α)% confidence interval is:
Home Page
Title Page
s
Cˆp
χ21−α/2,n−1
n−1
s
≤ Cp ≤ Cˆp
χ2α/2,n−1
n−1
JJ
II
J
I
Page 2 of 37
.
Go Back
Full Screen
Close
Quit
Example. Suppose U SL = 62, LSL =
38, n = 20, CL ≈ T = (U SL + LSL)/2
and S = 1.75, then the point estimate of
Home Page
Title Page
Cp is
JJ
II
J
I
Page 3 of 37
62
−
38
Cˆp =
= 2.29,
6 × 1.75
Go Back
Full Screen
Close
Quit
and the 95% confidence interval on Cp is
r
2.29
r
8.91
32.85
≤ Cp ≤ 2.29
,
19
19
1.57 ≤ Cp ≤ 3.01,
Home Page
Title Page
JJ
II
J
I
Page 4 of 37
where χ20.975,19
=
8.91 and χ20.025,19
= 32.85.
Go Back
Full Screen
Close
Quit
This confidence interval seems wider
(small sample).
• Use S rather than R/d2 to estimate σ and
Home Page
Title Page
the process must be in statistical in control.
Otherwise, S and R/d2 could be very dif-
JJ
II
J
I
Page 5 of 37
Go Back
ferent.
Full Screen
Close
Quit
• For Cpk and Cpm, the approx. 100(1 −
α)% confidence interval on Cpk is
Home Page
s
Cˆpk 1 − Zα/2
1
1
≤
C
+
pk
2
ˆ
9nCpk 2(n − 1)
s
1
1
ˆ
≤ Cpk 1 + Zα/2
.
+
2
9nCˆpk 2(n − 1)
Title Page
JJ
II
J
I
Page 6 of 37
Go Back
Full Screen
Close
Quit
Example. n = 20 from stable process
id used to estimate Cpk with Cˆpk = 1.33.
Home Page
Title Page
Then an approx. 95% confidence interval
on Cpk is
JJ
II
J
I
Page 7 of 37
Go Back
Full Screen
Close
Quit
r
1.33 1 − 1.96
1
1
+
2
9 × 20 × 1.33
2 × 19
≤ Cpk ≤
r
1.33 1 + 1.96
Home Page
Title Page
1
1
+
2
9 × 20 × 1.33
2 × 19
,
JJ
II
J
I
Page 8 of 37
Go Back
or 0.99 ≤ Cpk ≤ 1.67.
Full Screen
Close
Quit
It could be very wider since from 0.99 < 1
Home Page
(bad case ) to 1.67 (good case).
• Small sample is main reason.
Title Page
JJ
II
J
I
Page 9 of 37
Go Back
Full Screen
Close
Quit
• For Cpc, we estimate E|X − T | by c =
Pn
1
i=1 |xi − T |, it leads to the estimator
n
Home Page
Title Page
U SL − LSL
ˆ
p
Cpc =
.
6c π/2
JJ
II
J
I
Page 10 of 37
Go Back
Full Screen
Close
Quit
A 100(1 − α)% confidence interval for
E|X − T | is approx. given as follows:
Home Page
Title Page
sc
c ± tα/2,n−1 √ ,
n
JJ
II
J
I
Page 11 of 37
Go Back
Full Screen
Close
Quit
where
s2c
1
=
n−1
n
X
(|xi − T | − c)2
i=1
n
X
1
[
|xi − T |2 − nc2].
=
n − 1 i=1
Home Page
Title Page
JJ
II
J
I
Page 12 of 37
Go Back
Full Screen
Close
Quit
Thus, a 100(1 − α)% confidence interval
for Cpc is
Home Page
Title Page
Cˆpc
1+
tα/2,n−1sc
√
c n
≤ Cpc ≤
Cˆpc
1−
tα/2,n−1sc .
√
c n
JJ
II
J
I
Page 13 of 37
Go Back
Full Screen
Close
Quit
Testing hypothesis about PCRs
Consider the hypothesis testing problem:
Home Page
Title Page
H0 : Cp = Cp0(target value),
H1 : Cp > Cp0.
JJ
II
J
I
Page 14 of 37
Go Back
Full Screen
Close
Quit
Here we may define in advance Cp(High)
[Cp (Low)] as a process capability that we
Home Page
Title Page
would like to accept [reject] with prob. 1−
α [1 − β].
JJ
II
J
I
Page 15 of 37
Go Back
Full Screen
Close
Quit
Table
7-5
gives
Cp(H)/Cp(L)
and
Home Page
C/Cp(L) for varying sample sizes and
α = β = 0.05 or α = β = 0.10.
Title Page
JJ
II
J
I
Page 16 of 37
Go Back
Full Screen
Close
Quit
Home Page
Title Page
JJ
II
J
I
Page 17 of 37
Go Back
Full Screen
Close
Quit
See also the following figure. Actually,
q
C/Cp(L) = 1/ χ2α,n−1/(n − 1).
Home Page
Title Page
JJ
II
J
I
Page 18 of 37
Go Back
Full Screen
Close
Quit
Home Page
Title Page
JJ
II
J
I
Page 19 of 37
Go Back
Full Screen
Close
Quit
Home Page
Title Page
JJ
II
J
I
Page 20 of 37
Go Back
Full Screen
Close
Quit
Example. A supplier is told to demonstrate that his process capability exceeds
Cp = 1.33. Thus, he want to test
H0 : Cp = 1.33,
H1 : Cp > 1.33.
Home Page
Title Page
JJ
II
J
I
Page 21 of 37
Go Back
Full Screen
Close
Quit
If Cp is below 1.33, there will be a high
prob. of detecting this (90%), whereas if
Home Page
Title Page
Cp exceeds 1.66, there will be a high prob.
of judging the process capable (90%).
JJ
II
J
I
Page 22 of 37
Go Back
Full Screen
Close
Quit
It means α = 0.10, β = 0.10 and
Cp(H) = 1.66, Cp(L) = 1.33, Cp(H)/Cp(L) = 1.25.
Home Page
Form Table 7-5, we get n = 70 and
C/Cp(L) = 1.10 and C = 1.33 × 1.10 =
Title Page
JJ
II
J
I
Page 23 of 37
Go Back
1.46.
Full Screen
Close
Quit
Process performance indices
AIAG(Automotive
Industry
Action
Home Page
Group) recommended process capability
indices Cp and Cpk when the process is in
Title Page
JJ
II
J
I
Page 24 of 37
control with σˆ = R/d2.
Go Back
Full Screen
Close
Quit
When the process is not in control, AIAG
use process performance indices: Pp and
Ppk , Pˆp = (U SL − LSL)/(6S), Pˆpk is similar. S and R/d2 is quite different in this
Home Page
Title Page
JJ
II
J
I
Page 25 of 37
case.
Go Back
Full Screen
Close
Quit
Process capability analysis using a control chart
Home Page
• Histograms, PP, PCR summarize the performance of process. They do not address
Title Page
JJ
II
J
I
Page 26 of 37
the issue of statistical control.
Go Back
Full Screen
Close
Quit
• Control chart should be regard as the primary technique.
Home Page
Title Page
Bottle bursting strength. Sample size 5
for each subgroup.
JJ
II
J
I
Page 27 of 37
Go Back
Full Screen
Close
Quit
R chart:



U CL = D4R = 2.115 × 77.3 = 163.49,



CL = R = 77.3,




 LCL = D3R = 0.
Home Page
Title Page
JJ
II
J
I
Page 28 of 37
Go Back
Full Screen
Close
Quit
x chart:



U CL = x + A2R = 264.06 + 0.577 × 77.3






=
308.66,



CL = x = 264.06,





LCL = x − A2R = 264.06 − 0.577 × 77.3






=
219.46.
Home Page
Title Page
JJ
II
J
I
Page 29 of 37
Go Back
Full Screen
Close
Quit
µˆ = 264.06 and σˆ = R/d2 = 77.3/2.326 =
33.23.
Then, the one-side PCR is estimated by
Home Page
Title Page
µ − LSL 264.06 − 200
ˆ
=
= 0.64.
Cpl =
3ˆ
σ
3 × 33.23
JJ
II
J
I
Page 30 of 37
Go Back
This process capability is inadequate.
Full Screen
Close
Quit
• Although a process is in control, but operating at a an unacceptable level.
Home Page
• Control chart can be used as monitoring device or logbook or show the effect
Title Page
JJ
II
J
I
Page 31 of 37
of changes.
Go Back
Full Screen
Close
Quit
• Process is out-of-control, it is unsafe
to estimate the process capability. Stable
Home Page
process is needed.
Process capability analysis using de-
Title Page
JJ
II
J
I
Page 32 of 37
signed experiments
Go Back
Full Screen
Close
Quit
Control charts and tabular methods
•
2
σtotal
=
2
σproduct
+
2
σgage
.
Home Page
Title Page
JJ
II
J
I
Page 33 of 37
Go Back
Full Screen
Close
Quit
Measuring gage capability. See Figure
7-13. The std. of measurement error σgage
can be estimated by
σgage = R/d2 = 1.0/1.128 = 0.887.
Home Page
Title Page
JJ
II
J
I
Page 34 of 37
Go Back
Full Screen
Close
Quit
Home Page
Title Page
JJ
II
J
I
Page 35 of 37
Go Back
Full Screen
Close
Quit
Usually 6σgage is a good estimate of gage
capability:
6σgage = 6 × 0.887 = 5.32.
Home Page
Title Page
JJ
II
J
I
Page 36 of 37
Go Back
Full Screen
Close
Quit
• P/T ratio ( precision-to-tolerance band):
P
6ˆ
σgage
=
.
T
U SL − LSL
Home Page
Title Page
For example 7-7, we have P/T = 6 ×
JJ
II
J
I
0.887/(60 − 5) = 0.097.
Page 37 of 37
Go Back
Full Screen
Close
Quit