#### Concept of Central Limit Theorem

Central Limit Theorem:

Sampling Distribution of the Sample Means:

Rather than working with individual scores, statisticians frequently work with the means. What happens is that some samples are taken, the mean is evaluated for each sample and then the means are employed as the data, instead than individual scores being employed. The sample is a sampling distribution of sample means.

Whenever all the possible sample means are evaluated, then the given properties are true:

1) The mean of the sample means will be the mean of the population

2) The variance of sample means will be the variance of population divided by the sample size.

3) The standard deviation of the sample means (termed as the standard error of the mean) will be smaller than the population mean and will be equivalent to the standard deviation of population divided by the square root of sample size.

4) The sample means will encompass a normal distribution, whenever the population consists of a normal distribution, then the sample means will contain a normal distribution.

5) When the population is not normally distributed, however the sample size is adequately big, then the sample means will encompass an approximately normal distribution. Some of the books define adequately big as at least 30 and others as at least 31.

The formula for z-score whenever working with the sample means is as:

z = (x¯ - μ)/(σ/√n)

Finite Population Correction Factor:

When the sample size is greater than 5% of the population size and the sampling is completed without replacement, then a correction requires to be made to the standard error of means.

In the above formula, N is the population size and n is the sample size. The adjustment is to multiply standard error by the square root of quotient of the difference between the sample sizes and population and one less than the population size.

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