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7. Product and Process Comparisons
7.2. Comparisons based on data from one process

7.2.4.

Does the proportion of defectives meet requirements?

Testing proportion defective is based on the binomial distribution The proportion of defective items in a manufacturing process can be monitored using statistics based on the observed number of defectives in a random sample of size N from a continuous manufacturing process, or from a large population or lot. The proportion defective in a sample follows the binomial distribution where p is the probability of an individual item being found defective. Questions of interest for quality control are:
  1. Is the proportion of defective items within prescribed limits?
  2. Is the proportion of defective items less than a prescribed limit?
  3. Is the proportion of defective items greater than a prescribed limit?
Hypotheses regarding proportion defective The corresponding hypotheses that can be tested are:
  1. pp0

  2. p <= p0

  3. p >= p0
where p0 is the prescribed proportion defective.
Test statistic based on a normal approximation Given a random sample of measurements Y1, ..., YN from a population, the proportion of items that are judged defective from these N measurements is denoted phat. The test statistic

z = (phat - p(0))/SQRT(p(0)*(1 - p(0))/N)

depends on a normal approximation to the binomial distribution that is valid for large N, (N > 30). This approximation simplifies the calculations using critical values from the table of the normal distribution as shown below.

Restriction on sample size Because the test is approximate, N needs to be large for the test to be valid. One criterion is that N should be chosen so that

min{Np0, N(1 - p0)} ≥ 5

For example, if p0 = 0.1, then N should be at least 50 and if p0 = 0.01, then N should be at least 500. Criteria for choosing a sample size in order to guarantee detecting a change of size delta are discussed on another page.

One and two-sided tests for proportion defective Tests at the 1 - alpha confidence level corresponding to hypotheses (1), (2), and (3) are shown below. For hypothesis (1), the test statistic, z, is compared with z1-α/2, the critical value from the normal distribution that is exceeded with probability alpha/2 and similarly for (2) and (3). If

  1. z |   ≥   z1-α/2
  2. z   ≤   zα
  3. z   ≥   z1-α
the null hypothesis is rejected.
Example of a one-sided test for proportion defective After a new method of processing wafers was introduced into a fabrication process, two hundred wafers were tested, and twenty-six showed some type of defect. Thus, for N= 200, the proportion defective is estimated to be phat = 26/200 = 0.13. In the past, the fabrication process was capable of producing wafers with a proportion defective of at most 0.10. The issue is whether the new process has degraded the quality of the wafers. The relevant test is the one-sided test (3) which guards against an increase in proportion defective from its historical level.
Calculations for a one-sided test of proportion defective For a test at significance level alpha = 0.05, the hypothesis of no degradation is validated if the test statistic z is less than the critical value, z0.95 = 1.645. The test statistic is computed to be

z = (phat - p0)/SQRT{(p(0)*(1 - p(0))/N} =
 (.13 - .10)/SQRT{(.10*.90)/200} = 1.414
Interpretation Because the test statistic is less than the critical value (1.645), we cannot reject hypothesis (3) and, therefore, we cannot conclude that the new fabrication method is degrading the quality of the wafers. The new process may, indeed, be worse, but more evidence would be needed to reach that conclusion at the 95% confidence level.
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