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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Binomial proportion confidence interval ： ウィキペディア英語版 Binomial proportion confidence interval 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. Note that this does not mean that a calculated 95% confidence interval will contain the true proportion with 95% probability. Instead, one should interpret it as follows: the process of drawing a random sample and calculating an accompanying 95% confidence interval will generate a confidence interval that contains the true proportion in 95% of all cases. 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. ==Normal approximation interval== The most commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, $\backslash hat\; p$, with a normal distribution. However, although this distribution is frequently confused with a binomial distribution, it should be noted that the error distribution itself is not binomial, and hence other methods (below) are preferred. The approximation is usually justified by the central limit theorem. The formula is : $\backslash hat\; p\; \backslash pm\; z\; \backslash sqrt\backslash hat\; p\; \backslash left(1\; -\; \backslash hat\; p\; \backslash right)\}$ where $\backslash hat\; p$ is the proportion of successes in a Bernoulli trial process estimated from the statistical sample, $z$ is the $\backslash scriptstyle\; 1\; -\; \backslash frac\backslash alpha$ quantile of a standard normal distribution, $\backslash alpha$ is the error quantile and ''n'' is the sample size. For example, for a 95% confidence level the error ($\backslash alpha$) is 5%, so $\backslash scriptstyle\; 1\; -\; \backslash frac\backslash alpha$ = 0.975 and $z$ = 1.96. The central limit theorem applies poorly to this distribution with a sample size less than 30 or where the proportion is close to 0 or 1. The normal approximation fails totally when the sample proportion is exactly zero or exactly one. A frequently cited rule of thumb is that the normal approximation is a reasonable one as long as ''np'' > 5 and ''n''(1 − ''p'') > 5; see Brown et al. 2001.〔 〕 An important theoretical derivation of this confidence interval involves the inversion of a hypothesis test. Under this formulation, the confidence interval represents those values of the population parameter that would have large ''p''-values if they were tested as a hypothesized population proportion. The collection of values, $\backslash theta$, for which the normal approximation is valid can be represented as : $\backslash left\backslash \backslash hat\; p\; \backslash left(1\; -\; \backslash hat\; p\backslash right)\}\}\; \backslash le\; z\; \backslash right\backslash \}$ where $y$ is the $\backslash scriptstyle\; \backslash frac\backslash alpha$ quantile of a standard normal distribution. Since the test in the middle of the inequality is a Wald test, the normal approximation interval is sometimes called the Wald interval, but Pierre-Simon Laplace first described it in his 1812 book ''Théorie analytique des probabilités'' (page 283). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Binomial proportion confidence interval」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.