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CHAPTER 7 Tests of Hypotheses and Significance
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of H1: m 12. In case (iii), the P value 0.057 provides evidence for rejecting H0 in favor of H1: m 2 12 but not as much evidence as is provided for rejecting H0 in favor of H1: m 12. It should be kept in mind that the P value and the level of significance do not provide criteria for rejecting or not rejecting the null hypothesis by itself, but for rejecting or not rejecting the null hypothesis in favor of the alternative hypothesis. As the previous example illustrates, identical test results and significance levels can lead to different conclusions regarding the same null hypothesis in relation to different alternative hypotheses. When the test statistic S is the standard normal random variable, the table in Appendix C is sufficient to compute the P value, but when S is one of the t, F, or chi-square random variables, all of which have different distributions depending on their degrees of freedom, either computer software or more extensive tables than those in Appendices D, E, and F will be needed to compute the P value.
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Special Tests of Significance for Large Samples
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For large samples, many statistics S have nearly normal distributions with mean mS and standard deviation sS. In such cases we can use the above results to formulate decision rules or tests of hypotheses and significance. The following special cases are just a few of the statistics of practical interest. In each case the results hold for infinite populations or for sampling with replacement. For sampling without replacement from finite populations, the results must be modified. See pages 156 and 158. # 1. MEANS. Here S X, the sample mean; mS mX m, the population mean; sS sX s> !n, where s is the population standard deviation and n is the sample size. The standardized variable is given by Z # X m s> !n (1)
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s When necessary the observed sample standard deviation, s (or ^), is used to estimate s. To test the null hypothesis H0 that the population mean is m a, we would use the statistic (1). Then, if the alternative hypothesis is m 2 a, using a two-tailed test, we would accept H0 (or at least not reject it) at the 0.05 level if for a particular sample of size n having mean x # 1.96 x a # s> !n 1.96 (2)
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and would reject it otherwise. For other significance levels we would change (2) appropriately. To test H0 against the alternative hypothesis that the population mean is greater than a, we would use a one-tailed test and accept H0 (or at least not reject it) at the 0.05 level if x a # s> !n 1.645 (3)
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(see Table 7-1) and reject it otherwise. To test H0 against the alternative hypothesis that the population mean is less than a, we would accept H0 at the 0.05 level if x a # s> !n 1.645 (4)
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2. PROPORTIONS. Here S P, the proportion of successes in a sample; mS mP p, where p is the !pq>n, where q 1 p. The population proportion of successes and n is the sample size; sS sP standardized variable is given by Z In case P P p 2pq>n (5)
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X > n, where X is the actual number of successes in a sample, (5) becomes X np Z !npq
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Remarks similar to those made above about one- and two-tailed tests for means can be made.
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CHAPTER 7 Tests of Hypotheses and Significance
# # 3. DIFFERENCES OF MEANS. Let X1 and X2 be the sample means obtained in large samples of sizes n1 and n2 drawn from respective populations having means m1 and m2 and standard deviations s1 and s2. Consider the null hypothesis that there is no difference between the population means, i.e., m1 m2. From (11), page 157, on placing m1 m2 we see that the sampling distribution of differences in means is approximately normal with mean and standard deviation given by mX1 0 sX1 s2 1 A n1 s2 2 n2 (7)
s s where we can, if necessary, use the observed sample standard deviations s1 and s2 (or ^1 and ^2) as estimates of s1 and s2. By using the standardized variable given by Z # X1 # X2 sX1 X2 0 # # X1 X2 sX1 X2 (8)
in a manner similar to that described in Part 1 above, we can test the null hypothesis against alternative hypotheses (or the significance of an observed difference) at an appropriate level of significance. 4. DIFFERENCES OF PROPORTIONS. Let P1 and P2 be the sample proportions obtained in large samples of sizes n1 and n2 drawn from respective populations having proportions p1 and p2. Consider the null hypothesis that there is no difference between the population proportions, i.e., p1 p2, and thus that the samples are really drawn from the same population. From (13), page 157, on placing p1 p2 p, we see that the sampling distribution of differences in proportions is approximately normal with mean and standard deviation given by mP1 # where P 0 sP1 p(1 1 p)Q n 1 n2 R (9)
n1P1 n2P2 n1 n2 is used as an estimate of the population proportion p. By using the standardized variable Z P1 P2 0 sP1 P2 P1 P2 sP1 P2 (10)
we can test observed differences at an appropriate level of significance and thereby test the null hypothesis. Tests involving other statistics (see Table 5-1, page 160) can similarly be designed.
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