The Central Limit Theorem

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The central limit theorem states:

Ifis a random sample of sizedrawn from any population with meanand

variancethen the sample meanhas expected valueand expected variance If lots of samples, all of sizeare taken from the population, then the distribution of the sample means is approximately normally distributed,and the goodness of the fit improves with increasingThe underlying distribution of the population is arbitrary.

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

The central limit theorem can then be used to analyse the sample means, perfrom hypothesis testing and construct confidence intervals using the normal distribution.

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

probability that the sample mean is larger than 45.

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