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Theorem 5-5 is a consequence of the central limit theorem, page 112. It is assumed here that the population is infinite or that sampling is with replacement. Otherwise, the above is correct if we replace s> !n in (7) by sX as given by (6).
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Suppose that a population is infinite and binomially distributed, with p and q 1 p being the respective probabilities that any given member exhibits or does not exhibit a certain property. For example, the population may 1 be all possible tosses of a fair coin, in which the probability of the event heads is p 2. Consider all possible samples of size n drawn from this population, and for each sample determine the statistic that is the proportion P of successes. In the case of the coin, P would be the proportion of heads turning up in n tosses. Then we obtain a sampling distribution of proportions whose mean p and standard deviation p are given by mp p sp pq An p(1 A n p) (9)
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!pq. which can be obtained from (4) and (5) on placing m p, s For large values of n (n 30), the sampling distribution is very nearly a normal distribution, as is seen from Theorem 5-5. For finite populations in which sampling is without replacement, the second equation in (9) is replaced by sx as given by (6) with s !pq.
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CHAPTER 5 Sampling Theory
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Note that equations (9) are obtained most easily on dividing by n the mean and standard deviation (np and !npq) of the binomial distribution.
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Suppose that we are given two populations. For each sample of size n1 drawn from the first population, let us compute a statistic S1. This yields a sampling distribution for S1 whose mean and standard deviation we denote by mS1 and sS1, respectively. Similarly for each sample of size n2 drawn from the second population, let us compute a statistic S2 whose mean and standard deviation are mS2 and sS2, respectively. Taking all possible combinations of these samples from the two populations, we can obtain a distribution of the differences, S1 S2, which is called the sampling distribution of differences of the statistics. The mean and standard deviation of this sampling distribution, denoted respectively by mS1 S2 and sS1 S2, are given by mS1
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!s21 S
s22, S
(10)
provided that the samples chosen do not in any way depend on each other, i.e., the samples are independent (in other words, the random variables S1 and S2 are independent). # # If, for example, S1 and S2 are the sample means from two populations, denoted by X1, X2, respectively, then the sampling distribution of the differences of means is given for infinite populations with mean and standard deviation m1, s1 and m2, s2, respectively by mX1
2 2sX 1
2 sX 2
s2 1 A n1
s2 2 n2
(11)
using (4) and (5). This result also holds for finite populations if sampling is with replacement. The standardized variable Z # (X1 # X2) s2 1 A n1 (m1 s2 2 n2 m2) (12)
in that case is very nearly normally distributed if n1 and n2 are large (n1, n2 30). Similar results can be obtained for finite populations in which sampling is without replacement by using (4) and (6). Corresponding results can be obtained for sampling distributions of differences of proportions from two binomially distributed populations with parameters p1, q1 and p2, q2, respectively. In this case S1, and S2 correspond to the proportions of successes P1 and P2, and equations (11) yield mP1
!s2 1 P
s2 2 P
p1q1 A n1
p2q2 n2
(13)
Instead of taking differences of statistics, we sometimes are interested in the sum of statistics. In that case the sampling distribution of the sum of statistics S1 and S2 has mean and standard deviation given by mS1