## The Central Limit Theorem

The central limit theorem states:

Ifisa random sample of sizedrawnfrom any population with meanand

variancethenthe sample meanhasexpected valueandexpected varianceIflots of samples, all of sizearetaken from the population, then the distribution of the sample meansis approximately normally distributed,andthe goodness of the fit improves with increasingTheunderlying distribution of the population is arbitrary.

The central limit theorem is used for sampling when the samplesize is ‘large’, in practice over 50.

The central limit theorem can then be used to analyse the samplemeans, perfrom hypothesis testing and construct confidence intervalsusing the normal distribution.

A sample of size 100 is taken from a population with mean 40 andvariance 20. Find the

probability that the sample mean is larger than 45.

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