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469
CHAPTER 16
Section 16.1
1.
All ten values of the quality statistic are between the two control limits, so no outofcontrol
signal is generated.
2.
All ten values are between the two control limits.
However, it is readily verified that all but
one plotted point fall below the center line (at height .04975).
Thus even though no single
point generates an outofcontrol signal, taken together, the observed values do suggest that
there may be a decrease in the average value of the quality statistic.
Such a “small” change is
more easily detected by a CUSUM procedure (see section 16.5) than by an ordinary chart.
3.
P(10 successive points inside the limits) = P(1
st
inside) x P(2
nd
inside) x…x P(10
th
inside) =
(.998)
10
= .9802.
P(25 successive points inside the limits) = (.998)
25
= .9512.
(.998)
52
=
.9011, but (.998)
53
= .8993, so for 53 successive points the probability that at least one will
fall outside the control limits when the process is in control is 1  .8993 = .1007 > .10.
Section 16.2
4.
For Z, a standard normal random variable,
( 29
995
.
=
≤
≤

c
Z
c
P
implies that
( 29 ( 29
9975
.
2
005
.
995
.
=
+
=
≤
=
Φ
c
Z
P
c
.
Table A.3 then gives c = 2.81.
The
appropriate control limits are therefore
s
m
81
.
2
±
.
5.
a.
P(point falls outside the limits when
s
m
m
5
.
0
+
=
)
+
=
+
<
<


=
s
m
m
s
m
s
m
5
.
3
3
1
0
0
0
when
n
X
n
P
( 29
n
Z
n
P
5
.
3
5
.
3
1

<
<



=
( 29
0301
.
9699
.
1
882
.
1
12
.
4
1
=

=
<
<


=
Z
P
.
b.

=
+
<
<


s
m
m
s
m
s
m
0
0
0
3
3
1
when
n
X
n
P
( 29
n
Z
n
P
+
<
<
+


=
3
3
1
( 29
2236
.
24
.
5
76
.
1
=
<
<


=
Z
P
c.
( 29 ( 29
6808
.
47
.
1
47
.
7
1
2
3
2
3
1
=

<
<


=

<
<



Z
P
n
Z
n
P
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View Full Document Chapter 16:
Quality Control Methods
470
6.
The limits are
( 29( 29
80
.
00
.
13
5
6
.
3
00
.
13
±
=
±
, from which LCL = 12.20 and UCL = 13.80.
Every one of the 22
x
values is well within these limits, so the process appears to be in
control with respect to location.
7.
95
.
12
=
x
and
526
.
=
s
, so with
940
.
5
=
a
, the control limits are
70
.
13
,
20
.
12
75
.
95
.
12
5
940
.
526
.
3
95
.
12
=
±
=
±
.
Again, every point
( 29
x
is between
these limits, so there is no evidence of an outofcontrol process.
8.
336
.
1
=
r
and
325
.
2
5
=
b
, yielding the control limits
72
.
13
,
18
.
12
77
.
95
.
12
5
325
.
2
336
.
1
3
95
.
12
=
±
=
±
.
All points are between these limits,
so the process again appears to be in control with respect to location.
9.
54
.
96
24
07
.
2317
=
=
x
,
264
.
1
=
s
, and
952
.
6
=
a
, giving the control limits
17
.
98
,
91
.
94
63
.
1
54
.
96
6
952
.
264
.
1
3
54
.
96
=
±
=
±
.
The value of
x
on the 22
nd
day lies
above the UCL, so the process appears to be out of control at that time.
10.
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This note was uploaded on 03/30/2008 for the course STAT 211 taught by Professor Parzen during the Spring '07 term at Texas A&M.
 Spring '07
 Parzen

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