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# CLT - EC220 New version of a subsection of Section R.8...

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10.10.09. EC220 New version of a subsection of Section R.8 Central limit theorem If a random variable X has a normal distribution, its sample mean X will also have a normal distribution. This fact is useful for the construction of t statistics and confidence intervals if we are employing X as an estimator of the population mean. However, what happens if we are not able to assume that X is normally distributed? The standard response is to make use of a central limit theorem . Loosely speaking (we will make a more rigorous statement below), a central limit theorem states that the distribution of X will approximate a normal distribution as the sample size becomes large, even when the distribution of X is not normal. There are a number of central limit theorems, differing only in the assumptions that they make in order to obtain this result. Here we shall be content with using the simplest one, the Lindeberg–Levy central limit theorem. It states that, provided that the X i in the sample are all drawn independently from the same distribution (the distribution of X ), and provided that this distribution has finite population mean and variance, the distribution of X will converge on a normal distribution. This means that our t statistics and confidence intervals will be approximately valid after all, provided that the sample size is large enough. We will start by looking at two examples. Figure R.14 shows the distribution of X for the case where the X has a uniform distribution with range 0 to 1, for 10,000,000 samples. A uniform distribution is one in which all values over the range in question are equally likely.

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