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The following
is an excerpt from Chapter 3 of The
Quality Engineering Handbook by Thomas
Pyzdek, © Quality Publishing. It may be ordered from the Quality
Publishing Order Form.
Tolerance
intervals
We have found that confidence
limits may be determined so that the interval between these limits will
cover a population parameter with a certain confidence, that is, a certain
proportion of the time. Sometimes it is desirable to obtain an interval
which will cover a fixed portion of the population distribution with a
specified confidence. These intervals are called tolerance intervals,
and the end points of such intervals are called tolerance limits. For
example, a manufacturer may wish to estimate what proportion of product
will have dimensions that meet the engineering requirement. In quality
engineering, tolerance intervals are typically of the form , X-bar ± Ks,
where K is determined, so that the interval will cover a proportion P
of the population with confidence g. Confidence limits for m are also
of the form X-bar ± Ks. However, we determine k so that the confidence
interval would cover the population mean m a certain proportion of the
time. It is obvious that the interval must be longer to cover a large
portion of the distribution than to cover just the single value m. Table
11 in the Appendix gives K for P = 0.75, 0.90, 0.95, 0.99, 0.999 and g
= 0.75, 0.90, 0.95, 0.99 and for many different sample sizes n.
Example
of calculating a tolerance interval
Assume that a sample of
n=20 from a stable process produced the following results: . We can estimate
that the interval = 20 ± 3.615(1.5) = 20 ± 5.4225, or the interval from
14.5775 to 25.4225 will contain 99% of the population with confidence
95%. The K values in the table assume normally distributed populations.
Hypothesis
Testing
Statistical inference
generally involves four steps:
1. Formulating a hypothesis
about the population or "state of nature,"
2. Collecting a sample
of observations from the population,
3. Calculating statistics
based on the sample,
4. Either accepting or
rejecting the hypothesis based on a pre-determined acceptance criterion.
There are two types of
error associated with statistical inference
Type I error (α error)–The probability that
a hypothesis that is actually true will be rejected. The value of α (alpha) is known as the significance
level of the test.
Type II error (ß
error)–The probability that a hypothesis that is actually false will
be accepted.
Type II errors are often
plotted in what is known as an operating characteristics curve. Operating
characteristics curves will be used extensively in subsequent chapters
of this book in evaluating the properties of various statistical quality
control techniques.
Confidence intervals are
usually constructed as part of a statistical test of hypotheses. The hypothesis
test is designed to help us make an inference about the true population
value at a desired level of confidence. We will look at a few examples
of how hypothesis testing can be used in quality control applications.
Example:
hypothesis test of sample mean
Experiment: The nominal
specification for filling a bottle with a test chemical is 30 cc’s.
The plan is to draw a sample of n=25 units from a stable process and,
using the sample mean and standard deviation, construct a two-sided confidence
interval (an interval that extends on either side of the sample average)
that has a 95% probability of including the true population mean. If the
inter-val includes 30, conclude that the lot mean is 30, otherwise conclude
that the lot mean is not 30.
Result: A sample of 25
bottles was measured and the following statistics computed
The appropriate test statistic
is t, given by the formula
Table 6 in the Appendix
gives values for the t statistic at various degrees of freedom. There
are n-1 degrees of freedom. For our example we need the t.975 column and
the row for 24 df. This gives a t value of 2.064. Since the absolute value
of this t value is greater than our test statistic, we fail to reject
the hypothesis that the lot mean is 30 cc’s. Using statistical notation
this is shown as:
H0: m = 30
cc’s (the null hypothesis)
H1: m is not
equal to 30 cc’s (the alternate hypothesis)
a = .05 (type I error
or level of significance)
Critical region: -2.064
² t0 ² +2.064
Test statistic: t = -1.67.
Since t lies inside the
critical region, fail to reject H0, and accept the hypothesis
that the lot mean is 30cc for the data at hand.
Example:
hypothesis test of two sample variances
The variance of machine
X’s output, based on a sample of n = 25 taken from a stable process,
is 100. Machine Y’s variance, based on a sample of 10, is 50. The
manufacturing representative from the supplier of machine X contends that
the result is a mere "statistical fluke." Assuming that a "statistical
fluke" is something that has less than 1 chance in 100, test the
hypothesis that both variances are actually equal.
The test statistic used
to test for equality of two sample variances is the F statistic, which,
for this example, is given by the equation
Using Table 8 in the Appendix
for F.99 we find that for 24 df in the numerator and 9 df in
the denominator F = 4.73. Based on this we conclude that the manufacturer
of machine X could be right, the result could be a statistical fluke.
This example demonstrates the volatile nature of the sampling error of
sample variances and standard deviations.
Example:
hypothesis test of a standard deviation compared to a standard value
A machine is supposed
to produce parts in the range of 0.500 inches plus or minus 0.006 inches.
Based on this, your statistician computes that the absolute worst standard
deviation tolerable is 0.002 inches. In looking over your capability charts
you find that the best machine in the shop has a standard deviation of
0.0022, based on a sample of 25 units. In discussing the situation with
the statistician and management, it is agreed that the machine will be
used if a one-sided 95% confidence interval on sigma includes 0.002.
The correct statistic
for comparing a sample standard deviation with a standard value is the
chi-square statistic. For our data we have s=0.0022, n=25, and σ0=0.002. The Χ2 statistic has n-1 = 24
degrees of freedom. Thus,
Table 7 gives, in the
0.95 column (since we are constructing a one-sided confidence interval)
and the df = 24 row, the critical value c2 = 36.42. Since our computed
value of c2 is less than 36.42, we use the machine. The reader should
recognize that all of these exercises involved a number of assumptions.
E.g., that we "know" that the best machine has a standard deviation
of 0.0022. In reality, this knowledge must be confirmed by a stable control
chart.
Resampling
(Bootstrapping)
A number of criticisms
have been raised regarding the methods used for estimation and hypothesis
testing:
• They are not intuitive.
• They are based
on strong assumptions (e.g., normality) that are often not met in practice.
• They are difficult
to learn and to apply.
• They are error-prone.
In recent years a new
method of performing these analyses has been developed. It is known as
resampling or bootstrapping. The new methods are conceptually quite simple:
using the data from a sample, calculate the statistic of interest repeatedly
and examine the distribution of the statistic. For example, say you obtained
a sample of n=25 measurements from a lot and you wished to determine a
confidence interval on the statistic Cpk. Using resampling, you would
tell the computer to select a sample of n=25 from the sample results,
compute Cpk, and repeat the process many times, say 10,000 times. You
would then determine whatever percentage point value you wished by simply
looking at the results. The samples would be taken "with replacement,"
i.e., a particular value from the original sample might appear several
times (or not at all) in a resample.
Resampling has many advantages,
especially in the era of easily available, low-cost computer power. Spreadsheets
can be programmed to resample and calculate the statistics of interest.
Compared with traditional statistical methods, resampling is easier for
most people to understand. It works without strong assumptions, and it
is simple. Resampling doesn’t impose as much baggage between the
engineering problem and the statistical result as conventional methods.
It can also be used for more advanced problems, such as modeling, design
of experiments, etc.
For a discussion of the
theory behind resampling, see Efron (1982). For a presentation of numerous
examples using a resampling computer program see Simon (1992).
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