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2. Statistics > 2.3. Inferring Population Parameters from Sample Parameters

2.3. Inferring Population Parameters from Sample Parameters

Thus far, we have focused on statistics that describe a sample in various ways. A sample, however, is usually only a subset of the population. Given the statistics of a sample, what can we infer about the corresponding population parameters? If the sample is small or if the population is intrinsically highly variable, there is not much we can say about the population. However, if the sample is large, there is reason to hope that the sample statistics are a good approximation to the population parameters. We now quantify this intuition.

Our point of departure is the central limit theorem, which states that the sum of n independent random variables, for large n, is approximately normally distributed (see Section 1.7.5). Suppose that we collect a set of m samples, each with n elements, from some population. (In the rest of the discussion, we will assume that n is large enough that the central limit theorem applies.) If the elements of each sample are independently and randomly selected from the population, we can treat the sum of the elements of each sample as the sum of n independent and identically distributed random variables X1, X2,..., Xn. That is, the first element of the sample is the value assumed by the random variable X1, the second element is the value assumed by the random variable X2, and so on. From the central limit theorem, the sum of these random variables is normally distributed. The mean of each sample is the sum divided by a constant, so the mean of each sample is also normally distributed. This fact allows us to determine a range of values where, with high confidence, the population mean can be expected to lie.


  

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