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slides_12_inferfinite

# 1 it prevents us from looking at the data and then

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1. It prevents us from looking at the data and then deciding on the relevant alternative. 2. It is harder to reject a null against a two-sided alternative, forcing us to find even stronger evidence against H 0 before we reject it. 64

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We use the same test statistic, T X ̄ 0 / n n X ̄ 0 , but our rejection rule changes. We want to have power against 0 and 0 . We use a symmetric rule: Reject H 0 in favor of H 1 if | T | c for a suitable critical value c 0. Equivalently, reject H 0 if T c or T c . 65
As before, the choice of c is based based on the size of the test. If, for example, .05, we need to find the value c such that P | Z | c .05 where Z Normal 0,1 . Equivalently, P | Z | c .95 or P c Z c .95 or c c .95 66

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Now use the symmetry of the standard normal distribution: c 1 c . Plug in and solve to get c .975 or c 1 .975 1.96 67
For a two-sided alternative, the critical value for a size 5% test is now the 97.5 percentile in a standard normal distribution. The rejection rule, at the 5% size, is | T | 1.96 Compared with a one-sided alternative, we require a larger statistic in absolute value before we reject the null hypothesis: 1.96 compared with 1.65 (for H 1 : 0 ). A test against a two-sided alternative is called a two - tailed test . 68

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