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z 1 e !2p 3 ` Read QR In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Paint QR Code JIS X 0510 In None Using Barcode generation for Software Control to generate, create QR Code 2d barcode image in Software applications. u2>2 du
Quick Response Code Scanner In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. QR Code JIS X 0510 Creation In C#.NET Using Barcode printer for VS .NET Control to generate, create Quick Response Code image in .NET applications. Theorem 55 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). Generating QR Code In VS .NET Using Barcode drawer for ASP.NET Control to generate, create QRCode image in ASP.NET applications. QRCode Maker In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create QR Code JIS X 0510 image in .NET framework applications. Sampling Distribution of Proportions
Encoding QR Code In VB.NET Using Barcode creation for .NET framework Control to generate, create QR Code image in Visual Studio .NET applications. GS1  13 Encoder In None Using Barcode drawer for Software Control to generate, create GS1  13 image in Software applications. 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) Create Code 39 Extended In None Using Barcode printer for Software Control to generate, create Code39 image in Software applications. Barcode Maker In None Using Barcode drawer for Software Control to generate, create bar code image in Software applications. !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 55. 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. Creating UPCA Supplement 5 In None Using Barcode printer for Software Control to generate, create UPCA image in Software applications. Making EAN / UCC  13 In None Using Barcode creation for Software Control to generate, create GS1 128 image in Software applications. CHAPTER 5 Sampling Theory
Code11 Generator In None Using Barcode creation for Software Control to generate, create Code 11 image in Software applications. Barcode Scanner In .NET Using Barcode Control SDK for ASP.NET Control to generate, create, read, scan barcode image in ASP.NET applications. Note that equations (9) are obtained most easily on dividing by n the mean and standard deviation (np and !npq) of the binomial distribution. UPC  13 Maker In ObjectiveC Using Barcode generation for iPad Control to generate, create EAN 13 image in iPad applications. Drawing Code128 In None Using Barcode maker for Office Word Control to generate, create Code 128C image in Word applications. Sampling Distribution of Differences and Sums
Encoding Bar Code In ObjectiveC Using Barcode printer for iPhone Control to generate, create bar code image in iPhone applications. Decoding Data Matrix In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. 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 UPC A Scanner In C#.NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications. Painting Code39 In Java Using Barcode printer for BIRT reports Control to generate, create ANSI/AIM Code 39 image in Eclipse BIRT applications. !s21 S
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(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

