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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?

7.2.4.2.

Sample sizes required
Derivation of formula for required sample size when testing proportions The method of determining sample sizes for testing proportions is similar to the method for determining sample sizes for testing the mean. Although the sampling distribution for proportions actually follows a binomial distribution, the normal approximation is used for this derivation.
Minimum sample size If we are interested in detecting a change in the proportion defective of size delta in either direction, the minimum sample size is
  1. For a two-sided test

    N >= [p(1-p)/delta**2]*z(1-alpha/2)**2

  2. For a one-sided test

    N >= [p(1-p)/delta**2]*z(1-alpha)**2

Interpretation and sample size for high probability of detecting a change This requirement on the sample size only guarantees that a change of size delta is detected with 50% probability. The derivation of the sample size when we are interested in protecting against a change delta with probability 1 -beta (where beta is small) is
  1. For a two-sided test

    N >= (z(1-alpha/2) + z(1-beta))**2*[p(1-p)/delta**2]*z(1-alpha/2)**2

  2. For a one-sided test

    N >= (z(1-alpha) + z(1-beta))**2*[p(1-p)/delta**2]*z(alpha)**2

where z1-β is the critical value from the normal distribution that is exceeded with probability beta.
Value for the true proportion defective The equations above require that p be known. Usually, this is not the case. If we are interested in detecting a change relative to an historical or hypothesized value, this value is taken as the value of p for this purpose. Note that taking the value of the proportion defective to be 0.5 leads to the largest possible sample size.
Example of calculating sample size for testing proportion defective Suppose that a department manager needs to be able to detect any change above 0.10 in the current proportion defective of his product line, which is running at approximately 10% defective. He is interested in a one-sided test and does not want to stop the line except when the process has clearly degraded and, therefore, he chooses a significance level for the test of 5%. Suppose, also, that he is willing to take a risk of 10% of failing to detect a change of this magnitude. With these criteria:
  1. z0.95 = 1.645;   z0.90=1.282
  2. delta = 0.10
  3. p = 0.10
and the minimum sample size for a one-sided test procedure is

N >= [p(1-p)/delta**2]*[z(0.95)+z(0.90)]**2 = 0.10*0.90*2.927**2/0.10**2 is approximately 77

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