central_limit_apps

# central_limit_apps - Applications of the Central Limit...

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Applications of the Central Limit Theorem November 2, 2005 1 Introduction First, we state the central limit theorem Theorem 1 Suppose that X 1 , X 2 , ... is an infnite sequence oF independent, identically distributed random variables with common mean μ = E ( X 1 ) and fnite variance σ 2 = V ( x 1 ) . Then, iF we let S n = X 1 + ··· + X n we have that lim n →∞ P ± S n - σ n c ² = Φ( c ) = Z c -∞ e - x 2 / 2 2 π dx. There are many applications of this theorem to real-world problems, and in these notes we will give two: An application to hypothesis testing, and an application to noise cancellation. 2 Hypothesis Testing Here we will give an example of how to use the CLT to test hypotheses. We have already seen how to do this using a chi-square test to determine whether to reject a hypothesized population distribution (with ±nitely many classes) as being false. Here we will do this for when the population breaks down into two classes, smokers and non-smokers. 1

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2.1 Rejecting and Accepting Hypotheses Before we embark on the stated goal of this section, it is worth mentioning the diFerence between accepting and rejecting hypotheses, and what sorts of conclusions we can draw from statistical sampling. A primary misconception of certain statistical tests is that if a certain set of data supporting a hypothesis passes the test, then the hypothesis is likely true. This is the case of some statistical tests, but not of others. 2.2 The Main Problem Problem. You read in a newspaper that 20% of Georgians smoke, and you decide to test this hypothesis by doing a poll on 10 , 000 randomly selected Georgians with replacement (if the population you are testing is very large, then you would not need to test with replacement). Suppose that 2050 of the responses are “smoker”, while 7950 are “non-smoker”. Is the claim “20% of Georgians smoke” unreasonable? Well, in order to answer this question we would need more information; we would need to know what we mean by “unreasonable”. Here we will mean “unreasonable” with respect to a certain statistical test which we presently describe: Let X i = 1 if respondant i says he/she is a smoker, and let
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## central_limit_apps - Applications of the Central Limit...

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