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# Then the question is if so what is the probability of

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Unformatted text preview: Population Mean: Case 1 σ known Decision rule will evaluate how likely it is to get a sample with x = 2.5 if you population ¯ mean of 3 years. ¯ We know X ∼ N (3, (1.5)2 ) 25 if µ = 3 were true. ¯ Then the question is: If so what is the probability of getting an X &lt; 2.5? Well we can calculate that! 2.5 − 3 0.3 ¯ P (X &lt; 2.5) = P (Z &lt; ) = P (Z &lt; z ) = P (Z &lt; −1.66) z test statistic We can calculate this probability using the z-table. It is 0.0485: the probability that you get a sample that produces an x which is lower than than our current estimate x = 2.5 if the ¯ ¯ null hypothesis were true. We will call this probability p-value. This is a small probability, H0 should probably be rejected. But what is small enough? We need a criteria. We will set “a small enough probability”: signiﬁcance level and denote it with α. If you calculate a p-value which is less than α, you reject the null hypotesis. α is in our control: usually 0.1, 0.01 or 0.05. Remember α is also the probability of Type I error: we might be wrong! α is the probability associated the risk we are taking. This is the essence of p-value approach Utku Suleymanoglu (UMich) Hypothesis Testing 12 / 39 Testing Hypothesis about the Population Mean: Case 1 σ known Graphical recap: 0 Utku Suleymanoglu (UMich) Hypothesis Testing z 13 / 39 Testing Hypothesis about the Population Mean: Case 1 σ known p-value Approach for One-Tailed Tests We are working on the case: tests for µ where σ is known, but this logic generalizes to many cases. 1. After hypotheses statement and the calculation of test statistic (z for this case) : 2. Calculate Left-tailed tests: Calculate (left-tail) probability that the sample mean is less than x at hand if the ¯ null is true via: P (Z ≤ z ). Right-tailed tests: Calculate (right-tail) probability that the sample mean is more than x at hand if ¯ the null is true via: P (Z ≥ z ). 3. The probability you calculate is called the p-value. 4. Decision Rule: Comp...
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## This note was uploaded on 03/17/2014 for the course ECON 404 taught by Professor Staff during the Spring '08 term at University of Michigan.

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