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week04-3b - TA session 4 Econ 103 winter 2010 Wed Jan 27...

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TA session 4 Econ. 103, winter 2010 Wed., Jan. 27, 2010, 10:00 a.m. and 1:00 p.m. in PP2400E. 3 The equivalence of three hypothesis tests Suppose we want to test the hypothesis H 0 : μ = 1 against the two-sided alternative H 1 : μ 6 = 1 The sample mean ˆ μ = 1 n X i and sample variance ˆ s 2 = 1 n - 1 ( X i - ˆ μ ) 2 will be used to test this hypothesis. There are three tools available to test any hypothesis. These are the “t-test,” the “p-value” and the confidence interval. It is the mission of this sheet of paper to illustrate how they are equivalent. 3.1 Estimates Let’s say n = 30, ˆ μ = 1 . 2 and ˆ s 2 = 0 . 5. The question is whether the deviation of μ from our hypothesized value of “1” can be chalked up just to randomness. Such a judgment requires some statistical standard of “randomness.” 3.2 T ratio One standard is the t-ratio, which is a number that expresses the likelihood of drawing ˆ μ from a random variable distributed with mean of μ = 1 and whose standard deviation is estimated by ˆ s .
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