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


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:


Hypotheses regarding proportion defective 
The corresponding hypotheses that can be tested are:


Test statistic based on a normal approximation 
Given a random sample of measurements \(Y_1, \, \ldots, \, Y_N\)
from a population, the proportion of items that are judged
defective from these \(N\)
measurements is denoted \(\hat{p}\).
The test statistic,
$$ \large
z = \frac{\hat{p}  p_0}{\sqrt{\frac{p_0(1p_0)}{N}}} \, , $$


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 $$ \mbox{min }\{ N_{p_0}, \, N(1p_0)\} \ge 5 \, . $$ For example, if \(p_0\) = 0.1, then \(N\) should be at least 50 and if \(p_0\) = 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 twosided 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 \(z_{1\alpha/2}\),
the
critical value from
the normal distribution that is exceeded with probability \(\alpha/2\)
and similarly for (2) and (3). If


Example of a onesided test for proportion defective  After a new method of processing wafers was introduced into a fabrication process, two hundred wafers were tested, and twentysix showed some type of defect. Thus, for \(N\) = 200, the proportion defective is estimated to be \(\hat{p}\) = 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 onesided test (3) which guards against an increase in proportion defective from its historical level.  
Calculations for a onesided 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, \(z_{0.95}\)
= 1.645. The test statistic
is computed to be
$$ \large
z = \frac{\hat{p}  p_0}{\sqrt{\frac{p_0(1p_0)}{N}}}
= \frac{\hat{0.13}  0.10}{\sqrt{\frac{0.10(0.90)}{200}}}
= 1.414 \, . $$


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. 