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barcode font reporting services Tests of Hypotheses and Significance in Software
CHAPTER 7 Tests of Hypotheses and Significance Quick Response Code Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. QR Code JIS X 0510 Creation In None Using Barcode generator for Software Control to generate, create QR Code JIS X 0510 image in Software applications. EXAMPLE 7.1 If we obtain a sample of 100 tosses of a fair coin, so that n 100, p 2, then the expected frequency of heads (successes) is np (100)(1) 50. The observed frequency in the sample could of course be different. 2 QR Code 2d Barcode Decoder In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Quick Response Code Creator In C#.NET Using Barcode generator for VS .NET Control to generate, create Quick Response Code image in .NET applications. A natural generalization is to the case where k possible events A1, A2, c, Ak can occur, the respective probabilities being p1, p2, c, pk. In such cases, we have a multinomial population (see page 112). If we draw a sample of size n from this population, the observed frequencies for the events A1, c, Ak can be described by random variables X1, c, Xk (whose specific values x1, x2, c, xk would be the observed frequencies for the sample), while the expected frequencies would be given by np1, c, npk, respectively. The results can be indicated as in Table 72. Table 72 Event Observed Frequency Expected Frequency A1 x1 np1 A2 x2 np2 c c Ak xk npk QR Code 2d Barcode Generator In Visual Studio .NET Using Barcode maker for ASP.NET Control to generate, create QR Code image in ASP.NET applications. QR Code Maker In .NET Framework Using Barcode encoder for .NET Control to generate, create Quick Response Code image in .NET framework applications. 120, then the probabilities of the faces 1, EXAMPLE 7.2 If we obtain a sample of 120 tosses of a fair die, so that n 1 2, c, 6 are denoted by p1, p2, c, p6, respectively, and are all equal to 6. The corresponding expected frequencies are np1, np2, c, np6 and are all equal to (120) (1) 20. The observed frequencies of the various faces that come up in the 6 sample can of course be different. Generating QRCode In Visual Basic .NET Using Barcode printer for .NET Control to generate, create QR Code JIS X 0510 image in .NET framework applications. Bar Code Printer In None Using Barcode generator for Software Control to generate, create bar code image in Software applications. A clue as to the possible generalization of the statistic (6) which could measure the discrepancies existing between observed and expected frequencies in Table 72 is obtained by squaring the statistic (6) and writing it as Z2 (X np)2 npq (X1 np np)2 (X2 nq nq)2 (20) Data Matrix Drawer In None Using Barcode printer for Software Control to generate, create Data Matrix ECC200 image in Software applications. Generate EAN13 In None Using Barcode printer for Software Control to generate, create European Article Number 13 image in Software applications. where X1 X is the random variable associated with success and X2 n X1 is the random variable associated with failure. Note that nq in (20) is the expected frequency of failures. The form of the result (20) suggests that a measure of the discrepancy between observed and expected frequencies for the general case is supplied by the statistic x2 (X1 np1)2 np1 (X2 np2)2 np2 c (Xk npk)2 npk Encode Code 128A In None Using Barcode generator for Software Control to generate, create Code128 image in Software applications. Draw Code 39 Full ASCII In None Using Barcode generation for Software Control to generate, create Code39 image in Software applications. npj)2 npj
ISBN  13 Maker In None Using Barcode drawer for Software Control to generate, create ISBN  10 image in Software applications. Generate Universal Product Code Version A In ObjectiveC Using Barcode generation for iPad Control to generate, create Universal Product Code version A image in iPad applications. (21) EAN / UCC  13 Encoder In Java Using Barcode generator for BIRT Control to generate, create UCC.EAN  128 image in Eclipse BIRT applications. Creating Universal Product Code Version A In Visual Studio .NET Using Barcode creation for ASP.NET Control to generate, create UPCA image in ASP.NET applications. where the total frequency (i.e., the sample size) is n, so that X1 An expression equivalent to (21) is x2 X2 j a npj Data Matrix Scanner In VB.NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET framework applications. Printing EAN 13 In Java Using Barcode drawer for Android Control to generate, create EAN 13 image in Android applications. (22) Scan GS1  13 In VS .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET framework applications. Drawing UPC Code In None Using Barcode encoder for Font Control to generate, create UCC  12 image in Font applications. (23) If x2 0, the observed and expected frequencies agree exactly while if x2 0, they do not agree exactly. The larger the value of x2, the greater is the discrepancy between observed and expected frequencies. As is shown in Problem 7.62, the sampling distribution of x2 as defined by (21) is approximated very closely by the chisquare distribution [hence the choice of symbol in (21)] if the expected frequencies npj are at least equal to 5, the approximation improving for larger values. The number of degrees of freedom for this chisquare distribution is given by: (a) n k 1 if expected frequencies can be computed without having to estimate population parameters from sample statistics. Note that we subtract 1 from k because of the constraint condition (22), which states that if we know k 1 of the expected frequencies, the remaining frequency can be determined. (b) n k 1 m if the expected frequencies can be computed only by estimating m population parameters from sample statistics. CHAPTER 7 Tests of Hypotheses and Significance
In practice, expected frequencies are computed on the basis of a hypothesis H0. If under this hypothesis the computed value of x2 given by (21) or (23) is greater than some critical value (such as x2 or x2 , which are 0.95 0.99 the critical values at the 0.05 and 0.01 significance levels, respectively), we would conclude that observed frequencies differ significantly from expected frequencies and would reject H0 at the corresponding level of significance. Otherwise, we would accept it or at least not reject it. This procedure is called the chisquare test of hypotheses or significance. Besides applying to the multinomial distribution, the chisquare test can be used to determine how well other theoretical distributions, such as the normal or Poisson, fit empirical distributions, i.e., those obtained from sample data. See Problem 7.44.

