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Tests of Hypotheses and Significance
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If on the supposition that a particular hypothesis is true we find that results observed in a random sample differ markedly from those expected under the hypothesis on the basis of pure chance using sampling theory, we would say that the observed differences are significant and we would be inclined to reject the hypothesis (or at least not accept it on the basis of the evidence obtained). For example, if 20 tosses of a coin yield 16 heads, we would be inclined to reject the hypothesis that the coin is fair, although it is conceivable that we might be wrong. Procedures that enable us to decide whether to accept or reject hypotheses or to determine whether observed samples differ significantly from expected results are called tests of hypotheses, tests of significance, or decision rules.
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If we reject a hypothesis when it happens to be true, we say that a Type I error has been made. If, on the other hand, we accept a hypothesis when it should be rejected, we say that a Type II error has been made. In either case a wrong decision or error in judgment has occurred. In order for any tests of hypotheses or decision rules to be good, they must be designed so as to minimize errors of decision. This is not a simple matter since, for a given sample size, an attempt to decrease one type of error is accompanied in general by an increase in the other type of error. In practice one type of error may be more serious than the other, and so a compromise should be reached in favor of a limitation of the more serious error. The only way to reduce both types of error is to increase the sample size, which may or may not be possible.
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CHAPTER 7 Tests of Hypotheses and Significance
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In testing a given hypothesis, the maximum probability with which we would be willing to risk a Type I error is called the level of significance of the test. This probability is often specified before any samples are drawn, so that results obtained will not influence our decision. In practice a level of significance of 0.05 or 0.01 is customary, although other values are used. If for example a 0.05 or 5% level of significance is chosen in designing a test of a hypothesis, then there are about 5 chances in 100 that we would reject the hypothesis when it should be accepted, i.e., whenever the null hypotheses is true, we are about 95% confident that we would make the right decision. In such cases we say that the hypothesis has been rejected at a 0.05 level of significance, which means that we could be wrong with probability 0.05.
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To illustrate the ideas presented above, suppose that under a given hypothesis the sampling distribution of a statistic S is a normal distribution with mean mS and standard deviation sS. Also, suppose we decide to reject the hypothesis if S is either too small or too large. The distribution of the standardized variable Z (S mS )>sS is the standard normal distribution (mean 0, variance 1) shown in Fig. 7-1, and extreme values of Z would lead to the rejection of the hypothesis.
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As indicated in the figure, we can be 95% confident that, if the hypothesis is true, the z score of an actual sample statistic S will lie between 1.96 and 1.96 (since the area under the normal curve between these values is 0.95). However, if on choosing a single sample at random we find that the z score of its statistic lies outside the range 1.96 to 1.96, we would conclude that such an event could happen with the probability of only 0.05 (total shaded area in the figure) if the given hypothesis were true. We would then say that this z score differed significantly from what would be expected under the hypothesis, and we would be inclined to reject the hypothesis. The total shaded area 0.05 is the level of significance of the test. It represents the probability of our being wrong in rejecting the hypothesis, i.e., the probability of making a Type I error. Therefore, we say that the hypothesis is rejected at a 0.05 level of significance or that the z score of the given sample statistic is significant at a 0.05 level of significance. The set of z scores outside the range 1.96 to 1.96 constitutes what is called the critical region or region of rejection of the hypothesis or the region of significance. The set of z scores inside the range 1.96 to 1.96 could then be called the region of acceptance of the hypothesis or the region of nonsignificance. On the basis of the above remarks, we can formulate the following decision rule: (a) Reject the hypothesis at a 0.05 level of significance if the z score of the statistic S lies outside the range 1.96 to 1.96 (i.e., either z 1.96 or z 1.96). This is equivalent to saying that the observed sample statistic is significant at the 0.05 level. (b) Accept the hypothesis (or, if desired, make no decision at all) otherwise. It should be noted that other levels of significance could have been used. For example, if a 0.01 level were used we would replace 1.96 everywhere above by 2.58 (see Table 7-1). Table 6-1, page 196, can also be used since the sum of the level of significance and level of confidence is 100%.
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