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Chapter 7: Problem 27

Explain the central limit theorem.

Short Answer

Expert verified
The Central Limit Theorem (CLT) states that when a sufficiently large number of independent and identically distributed random variables are added, their mean will approximate a normal distribution, regardless of the shape of their original distribution.

Step by step solution

01

Definition

The Central Limit Theorem (CLT) is a statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. Furthermore, all the samples will follow an approximate normal distribution pattern, with all variances being approximately equal to the variance of the population divided by each sample's size.
02

Requirements for CLT to Hold

The theorem applies to independent and identically distributed random variables. This means that the elements in a sample occur independently, so the occurrence of one does not affect the occurrence of another. Furthermore, the term 'identically distributed' means that each variable has the same probability distribution as the others and the same parameters.
03

Example Illustration

Let's consider rolling a fair six-sided dice, the population mean is 3.5 and the population variance is 2.92. If we roll the dice 50 times (a sufficiently large sample size), the mean of the sums of these rolls (divided by 50) should be close to 3.5. Also the distribution of sums divided by 50 will approximate a normal distribution pattern.

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