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ssrs export to pdf barcode font Sampling Theory in Software
CHAPTER 5 Sampling Theory Quick Response Code Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Encode Denso QR Bar Code In None Using Barcode maker for Software Control to generate, create QR Code image in Software applications. Table 52 Heights of 100 Male Students at XYZ University
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Code 128 Code Set B Creation In None Using Barcode creator for Font Control to generate, create Code 128B image in Font applications. UPC Code Recognizer In C#.NET Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. If in Table 52 we recorded the relative frequency or percentage rather than the number of students in each class, the result would be a relative, or percentage, frequency distribution. For example, the relative or percentage frequency corresponding to the class 63 65 is 18>100, or 18%. The corresponding histogram is then similar to that in Fig. 51 except that the vertical axis is relative frequency instead of frequency. The sum of the rectangular areas is then 1, or 100%. We can consider a relative frequency distribution as a probability distribution in which probabilities are replaced by relative frequencies. Since relative frequencies can be thought of as empirical probabilities (see page 5), relative frequency distributions are known as empirical probability distributions. Create Linear In .NET Using Barcode generation for .NET framework Control to generate, create Linear image in VS .NET applications. GS1  13 Generation In Visual C# Using Barcode creation for .NET Control to generate, create EAN13 Supplement 5 image in .NET framework applications. Computation of Mean, Variance, and Moments for Grouped Data
Code39 Creation In None Using Barcode encoder for Font Control to generate, create Code 3/9 image in Font applications. Print EAN128 In .NET Using Barcode creator for Reporting Service Control to generate, create UCC.EAN  128 image in Reporting Service applications. We can represent a frequency distribution as in Table 53 by giving each class mark and the corresponding class frequency. The total frequency is n, i.e., c Table 53 Class Mark x1 x2 ( xk TOTAL Class Frequency f1 f2 ( fk n
Since there are f1 numbers equal to x1, f2 numbers equal to x2, . . . , fk numbers equal to xk, the mean is given by x # f1x1 f2x2 n c fk xk a fx n (24) CHAPTER 5 Sampling Theory
Similarly the variance is given by s2 f1(x1 x)2 # f2(x2 x)2 # n c fk(xk x)2 # a f (x n x)2 # (25) Note the analogy of (24) and (25) with the results (2), page 75, and (13), page 77, if fj >n correspond to empirical probabilities. In the case where class intervals all have equal size c, there are available short methods for computing the mean and variance. These are called coding methods and make use of the transformation from the class mark x to a corresponding integer u given by x a cu (26) 0. The coding formulas for the mean and vari where a is an arbitrarily chosen class mark corresponding to u ance are then given by x # s2 a c n a fu a fu2 n a cu # Similar formulas are available for higher moments. The r th moments about the mean and the origin, respectively, are given by f1(x1 x)r c fk(xk x)r # # # a f (x x)r mr (29) n n mr r f1xr 1 c n fk xr k a fxr n (30) c2 B
a fu n R
(27) c2(u2 # u2) # (28) The two kinds of moments are related by m1 m2 m3 m4 etc. If we write Mr a f (u n u)r # Mr r a f ur n 0 mr 2 mr 3 mr 4 mr2 1 3mr mr 1 2 4mr mr 1 3
2mr3 1 6mr2mr 1 2
(31) 3mr14
where u is given by (26), then the relations (31) also hold between the M s. But mr a f (x n x)r # a f [(a cu) n (a cu)]r # a fcr(u n u)r # crMr so that we obtain from (31) the coding formulas m1 m2 m3 m4 0 c2(Mr 2 c3(Mr 3 c4(Mr 4
Mr2) 1 3Mr Mr 1 2 4Mr Mr 1 3
2Mr3) 1 6Mr2Mr 1 2
(32) 3Mr4) 1 etc. The second equation of (32) is, of course, the same as (28). In a similar manner other statistics, such as skewness and kurtosis, can be found for grouped samples. CHAPTER 5 Sampling Theory
SOLVED PROBLEMS
Sampling distribution of means 5.1. A population consists of the five numbers 2, 3, 6, 8, 11. Consider all possible samples of size two which can be drawn with replacement from this population. Find (a) the mean of the population, (b) the standard deviation of the population, (c) the mean of the sampling distribution of means, (d) the standard deviation of the sampling distribution of means, i.e., the standard error of means. (a) (b) s2 and s (2 3.29. 6)2 (3 m 6)2 2 (6 5 3 6)2 6 5 (8 8 11 6)2 30 5 (11 6.0 6)2 16 9 0 5 4 25 10.8 (c) There are 5(5) 25 samples of size two which can be drawn with replacement (since any one of five numbers on the first draw can be associated with any one of the five numbers on the second draw). These are (2, 2) (3, 2) (6, 2) (8, 2) (11, 2) (2, 3) (3, 3) (6, 3) (8, 3) (11, 3) (2, 6) (3, 6) (6, 6) (8, 6) (11, 6) (2, 8) (3, 8) (6, 8) (8, 8) (11, 8) (2, 11) (3, 11) (6, 11) (8, 11) (11, 11) The corresponding sample means are 2.0 2.5 (1) 4.0 5.0 6.5 2.5 3.0 4.5 5.5 7.0 4.0 4.5 6.0 7.0 8.5 5.0 5.5 7.0 8.0 9.5 6.5 7.0 8.5 9.5 11.0 and the mean of the sampling distribution of means is mX illustrating the fact that mX sum of all sample means in (1) above 25 150 25 6.0

