Two-Tailed Test

Two-Sided Hypothesis Testing

In hypothesis testing, the hypotheses are always statements about a population parameter that partitions the set of possible values that the parameters might take. For example, letting μ be the parameter for which the hypothesis test is performed, a null hypothesis, which is referred to as H0, might be defined as

and its two-sided alternative hypothesis, which is referred to as H1, is defined as

The alternative hypothesis H1 does not make a statement about whether μ is greater than μ0 or less than μ0, which makes this a two-sided test. The difference between a one-sided test and a two-sided test lies solely in the specification of the alternative hypothesis. As a consequence, whereas a one-sided test specifies in its alternative hypothesis that the parameter is either greater than or less than the value specified in the null hypothesis (H1 is either μ > μ0 or μ < μ0), in a two-sided test, the direction of the alternative hypothesis is left unspecified.

Evidence for or against the null hypothesis is obtained by means of a test statistic, which is a function of the available data. Just as in the onesided case, in a two-sided hypothesis test the decision of whether to reject the null hypothesis H0 is based on a test statistic W(X) = W(X1,X2, …, XN) which is a function of a (random) sample X1, X2, …, XN of size N from the population under study. The test specifies a rejection rule that indicates in what situations H0 should be rejected. In a two-sided test, rejection occurs for both large and small values of W(X), whereas in a one-sided test, rejection occurs either for large or small values of the test statistic (but not both) as dictated by the alternative hypothesis. Formally, a two-sided rejection rule is defined as:

To establish the values of the critical values c1 and c2, it is common practice to follow the Neyman-Pearson approach and first choose a significance level α. The significance level α of the test is an upper bound to the probability of mistakenly rejecting H0 when H0 is true (probability of Type I error). Once the significance level has been fixed, the constants c1 and c2 are chosen so that the probability of rejecting H0 when H0 is true is (at most) equal to the significance level. In other words, c1 and c2 are chosen so

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