Chapter 21
21.1Chance
variation is caused by a number of randomly occurring events that are part of the production process
and that in general cannot be eliminated without changing the process.
21.3 Special variation is caused by specific events or factors that are frequently temporary and that can usually be
identified and eliminated
21.4a Chance variation represents the variation in student achievement caused by differences in preparation,
motivation, and ability.
b Special variation represents variation due to specific event or factors that can be corrected.
21.5
0124
.
)
9938
.
1
(
2
)
5
.
2

z
(
P
=

=
=
α
21.6 ARL =
0124
.
1
= 81
21.7
0456
.
)
9772
.
1
(
2
)
0
.
2

z
(
P
=

=
=
α
21.8 ARL =
0456
.
1
= 22
21.9 ARL =
0026
.
1
= 385
Number of units = Production
×
ARL = 100(385) = 38,500
21.10a From Betamean spreadsheet,
β
= .6603
b Probability = .6603
8
= .0361
21.11 P = 1 −
β
= 1 − .6603 = .3397; ARL =
3397
.
1
P
1
=
= 2.94
21.12 Number of units = Production
×
ARL = 50(385) = 19,250
21.13a From Betamean spreadsheet,
β
= .8133
b Probability = .8133
8
= .1914
193
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β
= 1 − .8133 = .1867; ARL =
1867
.
1
P
1
=
= 5.36
21.15 Sampling 3 units per hour means that on average we will produce 38,500 units before erroneously concluding
that the process is out of control when it isn’t.
Sampling 2 units per half hour reduces this figure by 50%. Sampling
4 units per hour means that when the process goes out of control, the probability of not detecting a shift of 1.5
standard deviations is .6603 and we will produce on average 2.94
×
100 = 294 units until the chart indicates a
problem. Sampling 2 units per half hour increases the probability of not detecting the shift to .8133 and decreases
the average number of units produced when the process is out of control to 50
×
5.36 = 268.
21.16 Number of units = Production
×
ARL = 2000(385) = 770,000
21.17a From Betamean spreadsheet,
β
= .7388
b Probability = .7388
4
= .2979
21.18 P = 1 −
β
= 1 − .7388 = .2612; ARL =
2612
.
1
P
1
=
= 3.83
21.19 Number of units = Production
×
ARL = 4000(385) = 1,540,000
21.20a From Betamean spreadsheet,
β
= .3659
b Probability = .3659
4
= .0179
21.21 P = 1 −
β
= 1 − .3659 = .6341; ARL =
6341
.
1
P
1
=
= 1.58
21.22 Sampling 10 units per half hour means that on average we will produce 770,000 units before erroneously
concluding that the process is out of control when it isn’t.
Sampling 20 units per hour doubles this figure. Sampling
10 units per half hour means that when the process goes out of control, the probability of not detecting a shift of .75
standard deviations is .7388 and we will produce on average 3.83
×
2000 = 7660 units until the chart indicates a
problem. Sampling 20 units per hour decreases the probability of not detecting the shift to .3659 and decreases the
average number of units produced when the process is out of control to 4000
×
1.58 = 6320.
21.23 Centerline =
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 Spring '10
 Kumar
 Prime number, ARL, control limit, Upper control limit

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