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One-Tailed and Two-Tailed Tests
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In the above test we displayed interest in extreme values of the statistic S or its corresponding z score on both sides of the mean, i.e., in both tails of the distribution. For this reason such tests are called two-tailed tests or two-sided tests.
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CHAPTER 7 Tests of Hypotheses and Significance
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Often, however, we may be interested only in extreme values to one side of the mean, i.e., in one tail of the distribution, as, for example, when we are testing the hypothesis that one process is better than another (which is different from testing whether one process is better or worse than the other). Such tests are called one-tailed tests or one-sided tests. In such cases the critical region is a region to one side of the distribution, with area equal to the level of significance. Table 7-1, which gives critical values of z for both one-tailed and two-tailed tests at various levels of significance, will be useful for reference purposes. Critical values of z for other levels of significance are found by use of the table of normal curve areas. Table 7-1 Level of Significance a Critical Values of z for One-Tailed Tests Critical Values of z for Two-Tailed Tests 0.10 1.28 or 1.28 1.645 and 1.645 0.05 1.645 or 1.645 1.96 and 1.96 0.01 2.33 or 2.33 2.58 and 2.58 0.005 2.58 or 2.58 2.81 and 2.81 0.002 2.88 or 2.88 3.08 and 3.08
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P Value
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In most of the tests we will consider, the null hypothesis H0 will be an assertion that a population parameter has a specific value, and the alternative hypothesis H1 will be one of the following assertions: (i) The parameter is greater than the stated value (right-tailed test). (ii) The parameter is less than the stated value (left-tailed test). (iii) The parameter is either greater than or less than the stated value (two-tailed test). In cases (i) and (ii), H1 has a single direction with respect to the parameter, and in case (iii), H1 is bidirectional. After the test has been performed and the test statistic S computed, the P value of the test is the probability that a value of S in the direction(s) of H1 and as extreme as the one that actually did occur would occur if H0 were true. For example, suppose the standard deviation s of a normal population is known to be 3, and H0 asserts that the mean m is equal to 12. A random sample of size 36 drawn from the population yields a sample mean x 12.95. The test statistic is chosen to be # Z # X 12 s> !n # X 12 , 0.5 12)>0.5 1.9. The
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which, if H0 is true, is the standard normal random variable. The test value of Z is (12.95 P value for the test then depends on the alternative hypothesis H1 as follows:
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(i) For H1: m 12 [case (i) above], the P value is the probability that a random sample of size 36 would yield a sample mean of 12.95 or more if the true mean were 12, i.e., P(Z 1.9) 0.029. In other words, the chances are about 3 in 100 that x 12.95 if m 12. # (ii) For H1: m 12 [case (ii) above], the P value of the test is the probability that a random sample of size 36 would yield a sample mean of 12.95 or less if the true mean were 12, i.e., P(Z 1.9) 0.97, or the chances are about 97 in 100 that x 12.95 if m 12. # (iii) For H1: m 2 12 [case (iii) above], the P value is the probability that a random sample of size 36 would yield a sample mean 0.95 or more units away from 12, i.e., x 12.95 or x 11.05, if the true mean # # 1.9) 0.057, which says the chances are about 6 in were 12. Here the P value is P(Z 1.9) P(Z 0.095 if m 12. 100 that Z x 12 u # Small P values provide evidence for rejecting the null hypothesis in favor of the alternative hypothesis, and large P values provide evidence for not rejecting the null hypothesis in favor of the alternative hypothesis. In case (i) of the above example, the small P value 0.029 is a fairly strong indicator that the population mean is greater than 12, whereas in case (ii), the large P value 0.97 strongly suggests that H0: m 12 should not be rejected in favor
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