Question

a) What is the Central Limit Theorem? It is always true that as the sample size,...

a) What is the Central Limit Theorem? It is always true that as the sample size, n, increases, the distribution of the sample means will be approximately normally distributed. Explain

b) If the underlying population of study is not normally distributed, how large should the sample size be? What if the population is normally distributed ?

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Answer #1

(a) The central limit theorem says that the mean of samples are normally distributed no matter what the distribution they are taken from.

Mathematically,

Let Y = X1 + X2 + X3 +.........

Define,

As n approaches infinity, Zn approaches the standard normal distribution.

By the definition, As n approaches infinity, Zn approaches the standard normal distribution. Hence it is always true (and in fact desirable) that as the sample size increases, the distribution of sample means will be approximately normally distributed.

(b) As a rough guideline, if the population is not normally distributed then it is suggested to have a sample size of at least 30 for a good approximation.

If the population is normally distributed then all the sample points would be on a normal distribution curve. So regardless of sample size, sample would be normally distributed.

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