1. No category

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
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confidence interval for PCR.
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• For the “first generation” ratio Cp, replace σ by S, Cˆp = (U SL − LSL)/(6S),
the 100(1 − α)% confidence interval is:
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s
Cˆp
χ21−α/2,n−1
n−1
s
≤ Cp ≤ Cˆp
χ2α/2,n−1
n−1
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.
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Example. Suppose U SL = 62, LSL =
38, n = 20, CL ≈ T = (U SL + LSL)/2
and S = 1.75, then the point estimate of
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Cp is
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62
−
38
Cˆp =
= 2.29,
6 × 1.75
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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,
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where χ20.975,19
=
8.91 and χ20.025,19
= 32.85.
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This confidence interval seems wider
(small sample).
• Use S rather than R/d2 to estimate σ and
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the process must be in statistical in control.
Otherwise, S and R/d2 could be very dif-
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• For Cpk and Cpm, the approx. 100(1 −
α)% confidence interval on Cpk is
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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)
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Example. n = 20 from stable process
id used to estimate Cpk with Cˆpk = 1.33.
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Then an approx. 95% confidence interval
on Cpk is
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r
1.33 1 − 1.96
1
1
+
2
9 × 20 × 1.33
2 × 19
≤ Cpk ≤
r
1.33 1 + 1.96
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1
1
+
2
9 × 20 × 1.33
2 × 19
,
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or 0.99 ≤ Cpk ≤ 1.67.
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It could be very wider since from 0.99 < 1
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(bad case ) to 1.67 (good case).
• Small sample is main reason.
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• For Cpc, we estimate E|X − T | by c =
Pn
1
i=1 |xi − T |, it leads to the estimator
n
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U SL − LSL
ˆ
p
Cpc =
.
6c π/2
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A 100(1 − α)% confidence interval for
E|X − T | is approx. given as follows:
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sc
c ± tα/2,n−1 √ ,
n
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where
s2c
1
=
n−1
n
X
(|xi − T | − c)2
i=1
n
X
1
[
|xi − T |2 − nc2].
=
n − 1 i=1
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Thus, a 100(1 − α)% confidence interval
for Cpc is
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Cˆpc
1+
tα/2,n−1sc
√
c n
≤ Cpc ≤
Cˆpc
1−
tα/2,n−1sc .
√
c n
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Testing hypothesis about PCRs
Consider the hypothesis testing problem:
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H0 : Cp = Cp0(target value),
H1 : Cp > Cp0.
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Here we may define in advance Cp(High)
[Cp (Low)] as a process capability that we
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would like to accept [reject] with prob. 1−
α [1 − β].
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Table
7-5
gives
Cp(H)/Cp(L)
and
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C/Cp(L) for varying sample sizes and
α = β = 0.05 or α = β = 0.10.
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See also the following figure. Actually,
q
C/Cp(L) = 1/ χ2α,n−1/(n − 1).
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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.
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If Cp is below 1.33, there will be a high
prob. of detecting this (90%), whereas if
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Cp exceeds 1.66, there will be a high prob.
of judging the process capable (90%).
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It means α = 0.10, β = 0.10 and
Cp(H) = 1.66, Cp(L) = 1.33, Cp(H)/Cp(L) = 1.25.
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Form Table 7-5, we get n = 70 and
C/Cp(L) = 1.10 and C = 1.33 × 1.10 =
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1.46.
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Process performance indices
AIAG(Automotive
Industry
Action
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Group) recommended process capability
indices Cp and Cpk when the process is in
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control with σˆ = R/d2.
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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
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case.
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Process capability analysis using a control chart
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• Histograms, PP, PCR summarize the performance of process. They do not address
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the issue of statistical control.
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• Control chart should be regard as the primary technique.
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Bottle bursting strength. Sample size 5
for each subgroup.
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R chart:



U CL = D4R = 2.115 × 77.3 = 163.49,



CL = R = 77.3,




 LCL = D3R = 0.
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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.
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µˆ = 264.06 and σˆ = R/d2 = 77.3/2.326 =
33.23.
Then, the one-side PCR is estimated by
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µ − LSL 264.06 − 200
ˆ
=
= 0.64.
Cpl =
3ˆ
σ
3 × 33.23
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This process capability is inadequate.
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• Although a process is in control, but operating at a an unacceptable level.
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• Control chart can be used as monitoring device or logbook or show the effect
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of changes.
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• Process is out-of-control, it is unsafe
to estimate the process capability. Stable
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process is needed.
Process capability analysis using de-
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signed experiments
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Control charts and tabular methods
•
2
σtotal
=
2
σproduct
+
2
σgage
.
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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.
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Usually 6σgage is a good estimate of gage
capability:
6σgage = 6 × 0.887 = 5.32.
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• P/T ratio ( precision-to-tolerance band):
P
6ˆ
σgage
=
.
T
U SL − LSL
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For example 7-7, we have P/T = 6 ×
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0.887/(60 − 5) = 0.097.
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