In statistics, a binomial proportion confidence interval is a confidence interval for a proportion in a statistical population. It uses the proportion estimated in a statistical sample and allows for sampling error. There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution. In general, a binomial distribution applies when an experiment is repeated a fixed number of times, each trial of the experiment has two possible outcomes (labeled arbitrarily success and failure), the probability of success is the same for each trial, and the trials are statistically independent.
A simple example of a binomial distribution is the set of various possible outcomes, and their probabilities, for the number of heads observed when a (not necessarily fair) coin is flipped ten times. The observed binomial proportion is the fraction of the flips which turn out to be heads. Given this observed proportion, the confidence interval for the true proportion innate in that coin is a range of possible proportions which may contain the true proportion. A 95% confidence interval for the proportion, for instance, will contain the true proportion 95% of the times that the procedure for constructing the confidence interval is employed.
There are several ways to compute a confidence interval for a binomial proportion. The normal approximation interval is the simplest formula, and the one introduced in most basic Statistics classes and textbooks. This formula, however, is based on an approximation that does not always work well. Several competing formulas are available that perform better, especially for situations with a small sample size and a proportion very close to zero or one. The choice of interval will depend on how important it is to use a simple and easy-to-explain interval versus the desire for better accuracy.
Read more about Binomial Proportion Confidence Interval: Normal Approximation Interval, Wilson Score Interval, Jeffreys Interval, Clopper-Pearson Interval, Agresti-Coull Interval, Special Cases, Comparison of Different Intervals
Famous quotes containing the words proportion, confidence and/or interval:
“The majority is never right. Never, I tell you! That’s one of these lies in society that no free and intelligent man can help rebelling against. Who are the people that make up the biggest proportion of the population—the intelligent ones or the fools? I think we can agree it’s the fools, no matter where you go in this world, it’s the fools that form the overwhelming majority.”
—Henrik Ibsen (1828–1906)
“If the child-caregiver relationship is nurturing, reliable and often even joyous, the child’s confidence in human relationships as a source of comfort and reciprocity will be strengthened and expanded in spite of the parent’s absence. The child will learn that not only are the parents to be trusted but that other people are trustworthy as well.”
—Alicia F. Lieberman (20th century)
“I was interested to see how a pioneer lived on this side of the country. His life is in some respects more adventurous than that of his brother in the West; for he contends with winter as well as the wilderness, and there is a greater interval of time at least between him and the army which is to follow. Here immigration is a tide which may ebb when it has swept away the pines; there it is not a tide, but an inundation, and roads and other improvements come steadily rushing after.”
—Henry David Thoreau (1817–1862)