7.
Product and Process Comparisons
7.2. Comparisons based on data from one process
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Testing defect densities is based on the Poisson distribution |
The number of defects observed in an area of size A units is
often assumed to have a Poisson
distribution with parameter A x D, where D is
the actual process defect density (D is defects per unit area).
In other words:
The questions of primary interest for quality control are:
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Normal approximation to the Poisson |
We assume that AD is large enough so that the normal
approximation to the Poisson applies (in other words, AD > 10
for a reasonable approximation and AD > 20 for a good one).
That translates to
where is the standard normal distribution function. |
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Test statistic based on a normal approximation |
If, for a sample of area A with a defect density target of
D0, a defect count of C is observed,
then the test statistic
can be used exactly as shown in the discussion of the test statistic for fraction defectives in the preceding section. |
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Testing the hypothesis that the process defect density is less than or equal to D0 |
For example, after choosing a sample size of area A (see below
for sample size calculation) we can reject that the process defect
density is less than or equal to the target D0
if the number of defects C in the sample is greater than
CA, where
and Z is the upper 100x(1-) percentile of the standard normal distribution. The test significance level is 100x(1-). For a 90% significance level use Z = 1.282 and for a 95% test use Z = 1.645. is the maximum risk that an acceptable process with a defect density at least as low as D0 "fails" the test. |
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Choice of sample size (or area) to examine for defects |
In order to determine a suitable area A
to examine for defects, you first need to choose an unacceptable defect
density level. Call this unacceptable defect density D1
= kD0, where k > 1.
We want to have a probability of less than or equal to is of "passing" the test (and not rejecting the hypothesis that the true level is D0 or better) when, in fact, the true defect level is D1 or worse. Typically will be .2, .1 or .05. Then we need to count defects in a sample size of area A, where A is equal to
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Example |
Suppose the target is D0 = 4 defects per wafer and
we want to verify a new process meets that target. We choose
= .1 to be the
chance of failing the test if the new process is as good as
D0
( = the Type I
error probability or the "producer's risk") and we choose
= .1 for the
chance of passing the test if the new process is as bad as 6 defects
per wafer (
= the Type II error probability or the "consumer's risk"). That means
Z = 1.282 and
Z1- = -1.282.
The sample size needed is A wafers, where
which we round up to 9. The test criteria is to "accept" that the new process meets target unless the number of defects in the sample of 9 wafers exceeds
In other words, the reject criteria for the test of the new process is 44 or more defects in the sample of 9 wafers. Note: Technically, all we can say if we run this test and end up not rejecting is that we do not have statistically significant evidence that the new process exceeds target. However, the way we chose the sample size for this test assures us we most likely would have had statistically significant evidence for rejection if the process had been as bad as 1.5 times the target. |