How do the Normal Distribution and the Central Limit Theorem

How do the Normal Distribution and the Central Limit Theorem relate? What is the connection between the sampling distribution of sample means, the normal distribution and the Z table?

Solution

Roughly, the central limit theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution.

When the sample size is greater than 30, then the sampling distribution of the sample means (for random sampling) follows a normal distribution irrespective of the distribution of the population. This is a consequence of the central limit theorem.

The Z table is used to estimate the interval of population mean or even to check the population mean using hypothesis test when the central limit theorem is satisfied.