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CHAPTER 5 Sampling Theory
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whose values can be represented by g(x1, . . . , xn ). The word statistic is often used for the random variable or for its values, the particular sense being clear from the context. In general, corresponding to each population parameter there will be a statistic to be computed from the sample. Usually the method for obtaining this statistic from the sample is similar to that for obtaining the parameter from a finite population, since a sample consists of a finite set of values. As we shall see, however, this may not always produce the best estimate, and one of the important problems of sampling theory is to decide how to form the proper sample statistic that will best estimate a given population parameter. Such problems are considered in later chapters. Where possible we shall try to use Greek letters, such as and , for values of population parameters, and Roman letters, m, s, etc., for values of corresponding sample statistics.
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Sampling Distributions
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As we have seen, a sample statistic that is computed from X1, . . . , Xn is a function of these random variables and is therefore itself a random variable. The probability distribution of a sample statistic is often called the sampling distribution of the statistic. Alternatively we can consider all possible samples of size n that can be drawn from the population, and for each sample we compute the statistic. In this manner we obtain the distribution of the statistic, which is its sampling distribution. For a sampling distribution, we can of course compute a mean, variance, standard deviation, moments, etc. The standard deviation is sometimes also called the standard error.
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The Sample Mean
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Let X1, X2, . . . , Xn denote the independent, identically distributed, random variables for a random sample of size n as described above. Then the mean of the sample or sample mean is a random variable defined by # X X1 X2 n c Xn (2)
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in analogy with (3), page 75. If x1, x2, . . . , xn denote values obtained in a particular sample of size n, then the mean for that sample is denoted by x #
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EXAMPLE 5.5
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x2 n
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If a sample of size 5 results in the sample values 7, 9, 1, 6, 2, then the sample mean is x # 7 9 1 5 6 2 5
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Sampling Distribution of Means
Let f (x) be the probability distribution of some given population from which we draw a sample of size n. Then # it is natural to look for the probability distribution of the sample statistic X, which is called the sampling distribution for the sample mean, or the sampling distribution of means. The following theorems are important in this connection. Theorem 5-1 The mean of the sampling distribution of means, denoted by mX , is given by # E(X) where is the mean of the population. mX m (4)
CHAPTER 5 Sampling Theory
Theorem 5-1 states that the expected value of the sample mean is the population mean. Theorem 5-2 If a population is infinite and the sampling is random or if the population is finite and sampling 2 is with replacement, then the variance of the sampling distribution of means, denoted by sX , is given by # E [(X where
m)2]
2 sX
s2 n
is the variance of the population. N,
Theorem 5-3 If the population is of size N, if sampling is without replacement, and if the sample size is n then (5) is replaced by
2 sX
s2 N n aN
n b 1
while mX is still given by (4). Note that (6) reduces to (5) as N S `. Theorem 5-4 If the population from which samples are taken is normally distributed with mean m and variance s2, then the sample mean is normally distributed with mean m and variance s2>n. Theorem 5-5 Suppose that the population from which samples are taken has a probability distribution with mean m and variance s2 that is not necessarily a normal distribution. Then the standardized vari# able associated with X, given by # X m s>!n
Z is asymptotically normal, i.e., lim P(Z nS` z)
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