# Confidence Interval for the Proportion

If a characteristic is present in a proportion of a population, **p**, the proportion of individuals with that characteristic, **p'**, in the samples of size **n** is distributed according to:

The confidence interval for the proportion is:

q = 1 − p.

The maximum error is:

#### Example

In an electronic components factory, the proportion of defective final components is 20%. After a series of investments designed to improve performance, a random sample of 500 components found that 90 of them were defective. What level of confidence should be taken to accept that performance has not changed?

p = 0.2 1 − p =0.8 p'= 90/ 500 = 0.18

E = 0.2 - 0.18 = 0.02

P (z_{α/2} > 1.12) = 1 − P (z_{α/2} ≤ 1.12) = 1 − 0.8686 = 0.1314

0.8686 - 0.1314 = 0.737

Level of confidence: 73.72%