7.2.4.1. Confidence intervals for large sample sizes

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

7.2.4.1. Confidence intervals for large sample sizes

Confidence intervals are another way of testing hypotheses

Confidence intervals that bracket the population proportion defective, p, are an alternative method for testing hypotheses regarding proportion defectives. Confidence intervals corresponding to the hypotheses tests on the previous page at a 100(1- alpha

)% confidence level are expressed as

Two-sided confidence interval for p
Lower one-sided confidence interval for p
Upper one-sided confidence interval for p

where phat is the estimate of the proportion defective from a sample of N; and z(alpha) is the upper critical value from the normal distribution that is exceeded with probability alpha .

Important restriction

These confidence intervals are approximate and are based on a normal approximation to the binomial distribution which is valid if N is large enough. We can use the same rule-of-thumb for determining whether or not N is sufficiently large as was used in preceding section.

Methods for constructing confidence intervals where N is small are shown on the next page. Criteria for choosing a sample size in order to guarantee detecting a change of size delta are discussed on another page.

Example of one-sided confidence interval

The 95% one-sided confidence interval for p₀ for the example from the previous page is:

p >= phat - SQRT(phat*(1-phat)/N)*z(alpha;
=> p >= 0.13 - SQRT(0.13*0.87/200)*1.654;
=> p >= 0.0907

Because the hypothesized value of p₀ = 0.10 is consistent with the confidence interval, the null hypothesis of no change in the level of proportion defective is accepted.

Correction factor for sampling from a finite population without replacement

Thus far, we have assumed that the sample comes from a continuous production line or a very large or infinite population. If, however, sampling is from a finite population and is done without replacement, a correction factor is applied to the confidence limits. The correction factor is

where N' is the number of units in the population. The corrected 100(1- alpha )% confidence intervals become

Two-sided confidence interval for p
Lower one-sided confidence interval for p
Upper one-sided confidence interval for p

If the population is extremely large, the correction factor will be close to one. As the sample size approaches the population size, the correction factor approaches zero.