Question

What is the central limit theorem and why is it important?

Answer 1

Statement of Central limit Theorem:

The central limit theorem states that if we have a population with mean $\mu$ and standard deviation $\sigma$ and take sufficiently large random samples from the population with replacement, then the distribution of the sample mean is asymptotically normal.

We can calculate the mean of the sample means for the random samples we choose from the population:

${\mu }_{\overline{X}}=\mu$

As well as the standard deviation of sample means:

${\sigma }_{\overline{X}}=\frac{\sigma }{\sqrt{n}}$

According to the central limit theorem, the form of the sampling distribution will approach normalcy as the sample size is sufficiently large (usually $n>30$). regardless of the population distribution.

Importance of Central Limit Theorem:

This is useful since the researcher never knows which mean in the sampling distribution corresponds to the population mean, but by taking numerous random samples from a population, the sample means will cluster together, allowing the researcher to obtain a very accurate estimate of the population mean.