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If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the probability distribution of these averages will closely approximate a normal distribution. The central limit theorem has several variants.
The table shown on the right can be used in a two-sample t-test to estimate the sample sizes of an experimental group and a control group that are of equal size, that is, the total number of individuals in the trial is twice that of the number given, and the desired significance level is 0.05. [4]
Usually the sample drawn has the same sample size as the original data. Then the estimate of original function F can be written as F ^ = F θ ^ {\displaystyle {\hat {F}}=F_{\hat {\theta }}} . This sampling process is repeated many times as for other bootstrap methods.
By the central limit theorem, sample means of moderately large samples are often well-approximated by a normal distribution even if the data are not normally distributed. However, the sample size required for the sample means to converge to normality depends on the skewness of the distribution of the original data.
The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example: The binomial distribution B ( n , p ) {\textstyle B(n,p)} is approximately normal with mean n p {\textstyle np} and variance n p ( 1 − p ) {\textstyle np(1-p)} for large n {\displaystyle n} and for p ...
The normal approximation depends on the de Moivre–Laplace theorem (the original, binomial-only version of the central limit theorem) and becomes unreliable when it violates the theorems' premises, as the sample size becomes small or the success probability grows close to either 0 or 1 . [4]
Galton box A Galton box demonstrated. The Galton board, also known as the Galton box or quincunx or bean machine (or incorrectly Dalton board), is a device invented by Francis Galton [1] to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximates a normal distribution.
Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a sum of n independent, identically distributed Bernoulli variables with parameter p. This fact is the basis of a hypothesis test, a "proportion z-test", for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic. [35]