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Rejection Region and Critical Values We have seen that we reject H0 when p < , (where p is the probability of our results or more extreme results, assumig H0 to be true). Our p value is gotten by taking our test statistic (e.g. Z Y ) and using its probability distribution (e.g. normal). However, there is another way to see if we should reject H0. For one tail (directional) tests Recall that we have defined Z as the Z value such that the probability (Z > Z) = . e.g. Z.05 = 1.645; Z.01 = 2.33, etc. Thus, whenever Z Y > Z, we know that p < and we can therefore reject H0 with a one-tail test with significance level . For two tail (non-directional) tests Z/2 is the Z value such that the probability (Z > Z/2) = /2. e.g. if = .05, /2 = .025 and Z/2 = 1.96 if = .01, /2 = .005 and Z/2 = 2.575 Thus, whenever Z Y > Z/2, we know that p < and we can therefore reject H0 with a two-tail test with significance level . Confidence Intervals and hypothesis testing A. Suppose our null hypothesis is that Y = 0. B. If 0 is outside the 1 - CI, we would reject that null hypothesis, if using a two tailed test with significance level . C. If 0 is within the 1 - CI, we would not reject that null hypothesis, if using a two tailed test with significance level . But even if 0 is within the confidence interval, this does not prove that = 0. Remember that 0 is not the only possible value of that will be within the confidence interval. The t-distribution Recall ZY Y 0 Y To compute our standard error we did not use the population standard deviation (Y) but rather used the sample estimate (sY). Gosset, known as Student, proved that when we do that the normal distribution is not quite the right distribution to use in computing the probabilities. He showed that the proper distribution is the t- distribution. The t-distribution is like the normal in also being a bell-curve but a) it is somewhat wider (has bigger tails) b) its exact form depends on degrees of freedom)