7.2.4.3. Sample sizes required

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

7.2.4.3. 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.

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

phat - delta <= p <= phat + delta

Relationship to confidence interval

For a (1-

)% confidence interval based on the normal distribution, where z(alpha/2

> is the upper critical value of the normal distribution which is exceeded with probability alpha/2

delta = SQRT(p*(1-p)/N)*z(alpha/2)

Minimum sample size

Thus, the minimum sample size is

For a two-sided interval

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

For a one-sided interval

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

is small) is

For a two-sided interval

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

For a one-sided interval

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

where

is the upper 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:

z_.05 = 1.645; z_.10=1.282
= 0.10
p = 0.10

and the minimum sample size for a one-sided test procedure is

N >= [p(1-p)/delta**2]*[z(.05)+z(.10)]**2 = 0.10*0.90*2.927**2/0.10**2
is approximately 77