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If we draw an object from an urn, we have the choice of replacing or not replacing the object into the urn before we draw again. In the first case a particular object can come up again and again, whereas in the second it can come up only once. Sampling where each member of a population may be chosen more than once is called sampling with replacement, while sampling where each member cannot be chosen more than once is called sampling without replacement. A finite population that is sampled with replacement can theoretically be considered infinite since samples of any size can be drawn without exhausting the population. For most practical purposes, sampling from a finite population that is very large can be considered as sampling from an infinite population.
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CHAPTER 5 Sampling Theory
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Clearly, the reliability of conclusions drawn concerning a population depends on whether the sample is properly chosen so as to represent the population sufficiently well, and one of the important problems of statistical inference is just how to choose a sample. One way to do this for finite populations is to make sure that each member of the population has the same chance of being in the sample, which is then often called a random sample. Random sampling can be accomplished for relatively small populations by drawing lots or, equivalently, by using a table of random numbers (Appendix H) specially constructed for such purposes. See Problem 5.43. Because inference from sample to population cannot be certain, we must use the language of probability in any statement of conclusions.
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A population is considered to be known when we know the probability distribution f (x) (probability function or density function) of the associated random variable X. For instance, in Example 5.1 if X is a random variable whose values are the heights (or weights) of the 12,000 students, then X has a probability distribution f (x). If, for example, X is normally distributed, we say that the population is normally distributed or that we have a normal population. Similarly, if X is binomially distributed, we say that the population is binomially distributed or that we have a binomial population. There will be certain quantities that appear in f (x), such as and in the case of the normal distribution or p in the case of the binomial distribution. Other quantities such as the median, moments, and skewness can then be determined in terms of these. All such quantities are often called population parameters. When we are given the population so that we know f (x), then the population parameters are also known. An important problem arises when the probability distribution f(x) of the population is not known precisely, although we may have some idea of, or at least be able to make some hypothesis concerning, the general behavior of f(x). For example, we may have some reason to suppose that a particular population is normally distributed. In that case we may not know one or both of the values and and so we might wish to draw statistical inferences about them.
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We can take random samples from the population and then use these samples to obtain values that serve to estimate and test hypotheses about the population parameters. By way of illustration, let us consider Example 5.1 where X is a random variable whose values are the various heights. To obtain a sample of size 100, we must first choose one individual at random from the population. This individual can have any one value, say, x1, of the various possible heights, and we can call x1 the value of a random variable X1, where the subscript 1 is used since it corresponds to the first individual chosen. Similarly, we choose the second individual for the sample, who can have any one of the values x2 of the possible heights, and x2 can be taken as the value of a random variable X2. We can continue this process up to X100 since the sample size is 100. For simplicity let us assume that the sampling is with replacement so that the same individual could conceivably be chosen more than once. In this case, since the sample size is much smaller than the population size, sampling without replacement would give practically the same results as sampling with replacement. In the general case a sample of size n would be described by the values x1, x2, . . . , xn of the random variables X1, X2, . . . , Xn. In the case of sampling with replacement, X1, X2, . . . , Xn would be independent, identically distributed random variables having probability distribution f(x). Their joint distribution would then be P(X1 x1, X2 x2, c, Xn xn) f (x1) f (x2) c f (xn) (1)
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Any quantity obtained from a sample for the purpose of estimating a population parameter is called a sample statistic, or briefly statistic. Mathematically, a sample statistic for a sample of size n can be defined as a function of the random variables X1, . . . , Xn, i.e., g(X1, . . . , Xn ). The function g(X1, . . . , Xn ) is another random variable,
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