7. Product and Process Comparisons
7.2. Comparisons based on data vrom one process
7.2.7. How can we determine whether the proportion of defectives produced by a process has changed from the "nominal" value?

7.2.7.2.

Sample sizes required

Derivation of formula for required sample size when testing proportions

The methods of determining sample sizes for proportions is similar to the methods use for sampling to estimate the normal mean. Although the sampling distribution for proportions actually follows a binomial distribution. The normal distribution can be used to approximate the binomial distribution when n is large.

The test statistic is

The sampling error, e , is equal to (- p) , the difference between sample proportion  and the parameter to be estimated (p). This sampling error can be defined as

Formula for required sample size

Steps to calculate required sample size

To calculate the sample size for proportions the following three quantities must be defined:

1. The level of confidence desired, 1 - a
2. The sampling error permitted, e
3. The population p 

Observe the following: items 2 and 3 are somewhat arbitrary. Often a is set to .10, but this is not mandatory. The sampling error is often defined as 5% of its true value. Again, this can be set to any desired percentage. 

The guess at p is based on some form of prior knowledge. (But really, how can we expect to state a value for the very item that we are taking a sample in order to determine it?) It turns out that if we take p = .5 we obtain the largest sample size possible. This implies the widest confidence intervals and the highest possible cost. Hence we achieve the maximum information, and the highest precision, but at a price. We spent more money and time.

How to test proportions

Let us conclude with an example. 

A department manager wished to to have 90% confidence of estimating the proportion of nonconformities in the product that he is responsible for. 

Not knowing the true value, he indicates that he wants to be within 4% of the true (but unknown) value. Lacking any further information about the true proportion he accepts as a first guess p = .5 (this is the worst case choice of p, leading to the largest required sample size). 

With these criteria on the table, we get:
Z = 1.645, e = .04, and p = 0.5. Then 

If this sample size is prohibitive for one or several reasons (too expensive, not enough manpower, not enough test equipment capacity, etc.), another calculation with changed parameters can be performed.

How to test proportions

The Finite Population Correction Factor

In order to determine the sample size when sampling without replacement, the finite population correction factor is applied as follows 

If the amount sampled exceeds about 25% of the population, the finite population correction factor will exert a substantial effect on the sample size.