The sign test is a very simple procedure for testing hypothe-
35.
Suppose we wish to test H
: the X and Y distributions are identical versus
H
a
: the X distribution is less spread out than the Y distribution The accompanying figure pictures X and Y distributions for
which H
a
is true. The Wilcoxon rank-sum test is not appro- priate in this situation because when H
a
is true as pictured, the Y’s will tend to be at the extreme ends of the combined
sample resulting in small and large Y ranks, so the sum of X
ranks will result in a W value that is neither large nor small. has the same probability, so
has the same distribution as does W when H
is true. Thus c can be chosen from
Appendix Table A.14 to yield a level a test. The accompany- ing data refers to medial muscle thickness for arterioles from
the lungs of children who died from sudden infant death syn- drome x’s and a control group of children y’s. Carry out
the test of H
versus H
a
at level .05.
SIDS
4.0 4.4
4.8 4.9
Control
3.7 4.1
4.3 5.1
5.6 Consult the Lehmann book in the chapter bibliography for
more information on this test, called the Siegel-Tukey test.
36.
The ranking procedure described in Exercise 35 is somewhat asymmetric, because the smallest observation receives rank
1, whereas the largest receives rank 2, and so on. Suppose both the smallest and the largest receive rank 1, the second
smallest and second largest receive rank 2, and so on, and let
be the sum of the X ranks. The null distribution of is
not identical to the null distribution of W, so different tables are needed. Consider the case
List all 35 possible orderings of the three X values among the seven
observations e.g., 1, 3, 7 or 4, 5, 6, assign ranks in the manner described, compute the value of
for each possi- bility, and then tabulate the null distribution of
. For the test that rejects if
, what value of c prescribes approx- imately a level .10 test? This is the Ansari-Bradley test; for
additional information, see the book by Hollander and Wolfe in the chapter bibliography.
w s c
W s
W s
m 5 3, n 5 4.
W s
W s
W r
Hollander, Myles, and Douglas Wolfe, Nonparametric Statistical Methods
2nd ed., Wiley, New York, 1999. A very good reference on distribution-free methods with an excel-
lent collection of tables. Lehmann, Erich, Nonparametrics: Statistical Methods Based on
Ranks, Springer, New York, 2006. An excellent discussion of
the most important distribution-free methods, presented with a great deal of insightful commentary.
X distribution
Y distribution
“Ranks”
:
1 3
5 6
4 2
. . .
Consider modifying the procedure for assigning ranks as fol- lows: After the combined sample of
observations is ordered, the smallest observation is given rank 1, the largest
observation is given rank 2, the second smallest is given rank 3, the second largest is given rank 4, and so on. Then if H
a
is true as pictured, the X values will tend to be in the middle of
the sample and thus receive large ranks. Let denote the
sum of the X ranks and consider rejecting H in favor of H
a
when . When
H is true, every possible set of X ranks
w r c
W r
m 1 n
Bibliography
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651
16
Quality Control Methods
INTRODUCTION
Quality characteristics of manufactured products have received much attention from design engineers and production personnel as well as from those con-
cerned with financial management. An article of faith over the years was that very high quality levels and economic well-being were incompatible goals.
Recently, however, it has become increasingly apparent that raising quality lev- els can lead to decreased costs, a greater degree of consumer satisfaction, and
thus increased profitability. This has resulted in renewed emphasis on statistical techniques for designing quality into products and for identifying quality prob-
lems at various stages of production and distribution. Control charting is now used extensively as a diagnostic technique for
monitoring production and service processes to identify instability and unusual circumstances. After an introduction to basic ideas in Section 16.1, a number
of different control charts are presented in the next four sections. The basis for most of these lies in our previous work concerning probability distributions of
various statistics such as the sample mean and sample proportion .
Another commonly encountered situation in industrial settings involves a decision by a customer as to whether a batch of items offered by a supplier is
of acceptable quality. In the last section of the chapter, we briefly survey some acceptance sampling methods for deciding, based on sample data, on the
disposition of a batch. Besides control charts and acceptance sampling plans, which were first
developed in the 1920s and 1930s, statisticians and engineers have recently introduced many new statistical methods for identifying types and levels of pro-
duction inputs that will ensure high-quality output. Japanese investigators, and pˆ 5 Xn
X
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in particular the engineerstatistician G. Taguchi and his disciples, have been very influential in this respect, and there is now a large body of material known
as “Taguchi methods.” The ideas of experimental design, and in particular frac- tional factorial experiments, are key ingredients. There is still much controversy
in the statistical community as to which designs and methods of analysis are best suited to the task at hand. The expository article by George Box et al., cited
in the chapter bibliography, gives an informative critique; the book by Thomas Ryan listed there is also a good source of information.
16.1
General Comments on Control Charts
A central message throughout this book has been the pervasiveness of naturally occurring variation associated with any characteristic or attribute of different indi-
viduals or objects. In a manufacturing context, no matter how carefully machines are calibrated, environmental factors are controlled, materials and other inputs are
monitored, and workers are trained, diameter will vary from bolt to bolt, some plastic sheets will be stronger than others, some circuit boards will be defective
whereas others are not, and so on. We might think of such natural random variation as uncontrollable background noise.
There are, however, other sources of variation that may have a pernicious impact on the quality of items produced by some process. Such variation may be
attributable to contaminated material, incorrect machine settings, unusual tool wear, and the like. These sources of variation have been termed assignable causes in the
quality control literature. Control charts provide a mechanism for recognizing situations where assignable causes may be adversely affecting product quality. Once
a chart indicates an out-of-control situation, an investigation can be launched to identify causes and take corrective action.
A basic element of control charting is that samples have been selected from the process of interest at a sequence of time points. Depending on the aspect of the process
under investigation, some statistic, such as the sample mean or sample proportion of defective items, is chosen. The value of this statistic is then calculated for each sample
in turn. A traditional control chart then results from plotting these calculated values over time, as illustrated in Figure 16.1.
Value of quality statistic
UCL Upper control limit Center
line
Time LCL Lower control limit
1 2 3 4 5 . . .
Figure 16.1 A prototypical control chart
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Notice that in addition to the plotted points themselves, the chart has a center line and two control limits. The basis for the choice of a center line is sometimes a
target value or design specification, for example, a desired value of the bearing diam- eter. In other cases, the height of the center line is estimated from the data. If the
points on the chart all lie between the two control limits, the process is deemed to be in control. That is, the process is believed to be operating in a stable fashion reflect-
ing only natural random variation. An out-of-control “signal” occurs whenever a plotted point falls outside the limits. This is assumed to be attributable to some
assignable cause, and a search for such causes commences. The limits are designed so that an in-control process generates very few false alarms, whereas a process not
in control quickly gives rise to a point outside the limits.
There is a strong analogy between the logic of control charting and our previ- ous work in hypothesis testing. The null hypothesis here is that the process is in con-
trol. When an in-control process yields a point outside the control limits an out-of-control signal, a type I error has occurred. On the other hand, a type II
error results when an out-of-control process produces a point inside the control limits. Appropriate choice of sample size and control limits the latter corresponding
to specifying a rejection region in hypothesis testing will make the associated error probabilities suitably small.
We emphasize that “in control” is not synonymous with “meets design speci- fications or tolerance.” The extent of natural variation may be such that the percent-
age of items not conforming to specification is much higher than can be tolerated. In such cases, a major restructuring of the process will be necessary to improve process
capability. An in-control process is simply one whose behavior with respect to vari- ation is stable over time, showing no indications of unusual extraneous causes.
Software for control charting is now widely available. The journal Quality Progress
contains many advertisements for statistical quality control computer pack- ages. In addition, SAS and Minitab, among other general-purpose packages, have
attractive quality control capabilities.
EXERCISES
Section 16.1 1–3
1.
A control chart for thickness of rolled-steel sheets is based on an upper control limit of .0520 in. and a lower limit of .0475 in.
The first ten values of the quality statistic in this case , the sample mean thickness of
sample sheets are .0506, .0493, .0502, .0501, .0512, .0498, .0485, .0500, .0505, and
.0483. Construct the initial part of the quality control chart, and comment on its appearance.
2.
Refer to Exercise 1 and suppose the ten most recent values of the quality statistic are .0493, .0485, .0490, .0503, .0492,
.0486, .0495, .0494, .0493, and .0488. Construct the relevant portion of the corresponding control chart, and comment on
its appearance.
3.
Suppose a control chart is constructed so that the probability of a point falling outside the control limits when the process
is actually in control is .002. What is the probability that ten successive points based on independently selected samples
will be within the control limits? What is the probability that 25 successive points will all lie within the control limits?
What is the smallest number of successive points plotted for n 5
5 X
which the probability of observing at least one outside the control limits exceeds .10?
4.
A cork intended for use in a wine bottle is considered accept- able if its diameter is between 2.9 cm and 3.1 cm so the
lower specification limit is and
the upper specifi-
cation limit is .
a.
If cork diameter is a normally distributed variable with mean value 3.04 cm and standard deviation .02 cm, what
is the probability that a randomly selected cork will con- form to specification?
b.
If instead the mean value is 3.00 and the standard devia- tion is .05, is the probability of conforming to specifica-
tion smaller or larger than it was in a?
5.
If a process variable is normally distributed, in the long run virtually all observed values should be between
and , giving a process spread of 6s.
a.
With LSL and USL denoting the lower and upper specifi- cation limits, one commonly used process capability
index is .
The value
C
p
5 1
C
p
5 USL 2 LSL6s
m 1 3s
m 2 3s
USL 5 3.1 LSL 5 2.9
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indicates a process that is only marginally capable of meeting specifications. Ideally, C
p
should exceed 1.33 a “very good” process. Calculate the value of C
p
for each of the cork production processes described in the previous
exercise, and comment.
b.
The C
p
index described in a does not take into account process location. A capability measure that does involve
the process mean is Calculate the value of C
pk
for each of the cork-production processes described in the previous exercise, and com-
ment. [Note: In practice, m and s have to be estimated from process data; we show how to do this in Section 16.2]
c.
How do C
p
and C
pk
compare, and when are they equal? C
pk
5 min
5USL 2 m3s, m 2 LSL3s6
16.2
Control Charts for Process Location
Suppose the quality characteristic of interest is associated with a variable whose observed values result from making measurements. For example, the characteristic
might be resistance of electrical wire ohms, internal diameter of molded rubber expansion joints cm, or hardness of a certain alloy Brinell units. One important
use of control charts is to see whether some measure of location of the variable’s distribution remains stable over time. The most popular chart for this purpose is the
chart.
The X Chart Based on Known Parameter Values
Because there is uncertainty about the value of the variable for any particular item or specimen, we denote such a random variable rv by X. Assume that for an in-
control process, X has a normal distribution with mean value m and standard devi- ation s. Then if denotes the sample mean for a random sample of size n selected
at a particular time point, we know that