Lesson 27: The Central Limit Theorem

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quincunxIntroduction

In the previous lesson, we investigated the probability distribution ("sampling distribution") of the sample mean when the random sample X1, X2, ..., Xn comes from a normal population with mean μ and variance σ2, that is, when Xi ~ N(μ, σ2), i = 1, 2, ..., n. Specifically, we learned that if Xi, i = 1, 2,... , n, is a random sample of size n from a N(μ, σ2) population, then:

 \(\bar{X}\sim N\left(\mu,\dfrac{\sigma^2}{n}\right)\)

But what happens if the Xi follow some other non-normal distribution?  For example, what distribution does the sample mean follow if the Xi come from the Uniform(0, 1) distribution? Or, what distribution does the sample mean follow if the Xcome from a chi-square distribution with three degrees of freedom? Those are the kinds of questions we'll investigate in this lesson. As the title of this lesson suggests, it is the Central Limit Theorem that will give us the answer.

Objectives

  • To learn the Central Limit Theorem.
  • To get an intuitive feeling for the Central Limit Theorem.
  • To use the Central Limit Theorem to find probabilities concerning the sample mean.
  • To be able to apply the methods learned in this lesson to new problems.