barcode font reporting services Tests of Hypotheses and Significance in Software

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CHAPTER 7 Tests of Hypotheses and Significance
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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
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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 7-2. Table 7-2 Event Observed Frequency Expected Frequency A1 x1 np1 A2 x2 np2 c c Ak xk npk
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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.
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A clue as to the possible generalization of the statistic (6) which could measure the discrepancies existing between observed and expected frequencies in Table 7-2 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)
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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
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npj)2 npj
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(21)
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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
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(22)
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(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 chi-square 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 chi-square 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 chi-square test of hypotheses or significance. Besides applying to the multinomial distribution, the chi-square 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.
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