7.
Product and Process Comparisons
7.2. Comparisons based on data from one process 7.2.4. Does the proportion of defectives meet requirements?


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.  
Problem formulation 
We want to test the hypothesis
\(H_a: \,\, p \ne p_0\) Define \(\delta\) as the change in the proportion defective that we are interested in detecting
\(P(\mbox{reject } H_0  H_0 \mbox{ is false with any } p \le \delta) \le 1\beta\). 

Definition of allowable deviation  If we are interested in detecting a change in the proportion defective of size \(\delta\) in either direction, the corresponding confidence interval for \(p\) can be written as $$ \hat{p}  \delta \le p \le \hat{p} + \delta \, . $$  
Relationship to confidence interval  For a \(100(1\alpha)\) % confidence interval based on the normal distribution, where \(z_{1  \alpha/2}\) is the critical value of the normal distribution which is exceeded with probability \(\alpha/2\), is $$ \delta = z_{1\alpha/2} \, \sqrt{\frac{p_0 (1p_0)}{N}} + z_{1\beta} \, \sqrt{\frac{p_1 (1p_1)}{N}} \, . $$  
Minimum sample size 
Thus, the minimum sample size is


Continuity correction 
Fleiss, Levin, and Paik also recommend the following continuity
correction,


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 onesided 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:
