552 Chapter 13:Quality Control

with

(

1

2

)

=

∫∞

0

e−xx−1/2dx

=

∫∞

0

e−y

(^2) /2
√
2
y
ydy byx=
y^2
2
dx=ydy

√
2
∫∞
0
e−y
(^2) /2
dy
= 2
√
π
1
√
2 π
∫∞
0
e−y
(^2) /2
dy
= 2
√
πP[N(0, 1)> 0 ]

√
π
The preceding estimates forμandσmake use of allksubgroups and thus are reasonable
only if the process has remained in control throughout. To check this, we compute the
control limits based on these estimates ofμandσ, namely,
LCL=X−
3 S
√
nc(n)
(13.2.4)
UCL=X+
3 S
√
nc(n)
We now check that each of the subgroup averagesXifalls within these lower and upper
limits. Any subgroup whose average value does not fall within the limits is removed (we
suppose that the process was temporarily out of control) and the estimates are recomputed.
We then again check that all the remaining subgroup averages fall within the control limits.
If not, then they are removed, and so on. Of course, if too many of the subgroup averages
fall outside the control limits, then it is clear that no control has yet been established.
EXAMPLE 13.2b Let us reconsider Example 13.2a under the new supposition that the
process is just beginning and soμandσare unknown. Also suppose that the sample
standard deviations were as follows:
X S X S
1 3.01 .12 6 3.02 .08
2 2.97 .14 7 3.10 .15
3 3.12 .08 8 3.14 .16
4 2.99 .11 9 3.09 .13
5 3.03 .09 10 3.20 .16