Confidence Intervals For Proportions and Related Quantities
See also: Margin of error and Binomial proportion confidence intervalAn approximate confidence interval for a population mean can be constructed for random variables that are not normally distributed in the population, relying on the central limit theorem, if the sample sizes and counts are big enough. The formulae are identical to the case above (where the sample mean is actually normally distributed about the population mean). The approximation will be quite good with only a few dozen observations in the sample if the probability distribution of the random variable is not too different from the normal distribution (e.g. its cumulative distribution function does not have any discontinuities and its skewness is moderate).
One type of sample mean is the mean of an indicator variable, which takes on the value 1 for true and the value 0 for false. The mean of such a variable is equal to the proportion that have the variable equal to one (both in the population and in any sample). This is a useful property of indicator variables, especially for hypothesis testing. To apply the central limit theorem, one must use a large enough sample. A rough rule of thumb is that one should see at least 5 cases in which the indicator is 1 and at least 5 in which it is 0. Confidence intervals constructed using the above formulae may include negative numbers or numbers greater than 1, but proportions obviously cannot be negative or exceed 1. Additionally, sample proportions can only take on a finite number of values, so the central limit theorem and the normal distribution are not the best tools for building a confidence interval. See "Binomial proportion confidence interval" for better methods which are specific to this case.
Read more about this topic: Confidence Interval
Famous quotes containing the words confidence, intervals, proportions, related and/or quantities:
“If the child-caregiver relationship is nurturing, reliable and often even joyous, the childs confidence in human relationships as a source of comfort and reciprocity will be strengthened and expanded in spite of the parents 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)
“We say that every man is entitled to be valued by his best moment. We measure our friends so. We know, they have intervals of folly, whereof we take no heed, but wait the reappearings of the genius, which are sure and beautiful.”
—Ralph Waldo Emerson (18031882)
“I see every thing I paint in this world, but everybody does not see alike. To the eyes of a miser a guinea is more beautiful than the sun, and a bag worn with the use of money has more beautiful proportions than a vine filled with grapes.”
—William Blake (17571827)
“One does not realize the historical sensation as a re-experiencing, but as an understanding that is closely related to the understanding of music, or rather of the world by means of music.”
—Johan Huizinga (18721945)
“James Brown and Frank Sinatra are two different quantities in the universe. They represent two different experiences of the world.”
—Imamu Amiri Baraka [Everett Leroi Jones] (b. 1934)