 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
ssrs 2008 r2 barcode font Sampling Theory in Software
CHAPTER 5 Sampling Theory Scan QR Code ISO/IEC18004 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Encoding QR In None Using Barcode printer for Software Control to generate, create Quick Response Code image in Software applications. 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. Quick Response Code Recognizer In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Denso QR Bar Code Creation In C# Using Barcode creator for Visual Studio .NET Control to generate, create QR Code image in .NET applications. Sampling Distributions
Drawing QR Code 2d Barcode In .NET Using Barcode maker for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. Encode QR Code ISO/IEC18004 In VS .NET Using Barcode creator for Visual Studio .NET Control to generate, create QRCode image in .NET applications. 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. Draw QR In VB.NET Using Barcode creation for .NET framework Control to generate, create QR Code 2d barcode image in .NET framework applications. Make Barcode In None Using Barcode drawer for Software Control to generate, create bar code image in Software applications. The Sample Mean
Bar Code Drawer In None Using Barcode printer for Software Control to generate, create bar code image in Software applications. Drawing GS1128 In None Using Barcode maker for Software Control to generate, create UCC128 image in Software applications. 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) Create Code 128 Code Set A In None Using Barcode printer for Software Control to generate, create Code128 image in Software applications. Making EAN13 In None Using Barcode maker for Software Control to generate, create European Article Number 13 image in Software applications. 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 # MSI Plessey Printer In None Using Barcode encoder for Software Control to generate, create MSI Plessey image in Software applications. EAN / UCC  14 Maker In Java Using Barcode creator for Android Control to generate, create EAN / UCC  13 image in Android applications. EXAMPLE 5.5
Matrix Barcode Encoder In .NET Framework Using Barcode encoder for ASP.NET Control to generate, create Matrix Barcode image in ASP.NET applications. Creating Bar Code In ObjectiveC Using Barcode creation for iPad Control to generate, create barcode image in iPad applications. x2 n
Scanning Code 128 Code Set A In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Barcode Scanner In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. 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 Linear 1D Barcode Generator In Visual Studio .NET Using Barcode encoder for .NET framework Control to generate, create Linear Barcode image in .NET applications. Painting EAN13 In .NET Using Barcode generation for ASP.NET Control to generate, create EAN 13 image in ASP.NET applications. 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 51 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 51 states that the expected value of the sample mean is the population mean. Theorem 52 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 53 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 54 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 55 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)

