Then the question is if so what is the probability of

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Unformatted text preview: about the Population Mean: 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. .5 2 ¯ We know X ∼ N (3, (125) ) if µ = 3 were true. ¯ Then the question is: If so what is the probability of getting an X ≤ 2.5? Well we can calculate that! ¯ P ( X < 2 .5 ) = P ( Z < 2.5 − 3 ) = P (Z < z ) = P (Z < −1.66) 0 .3 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 equal to or lower than than our ¯ current estimate x = 2.5 if the null hypothesis were true. ¯ This is a small probability, so we should probably think that H0 is false. But what is a small probability? We need a criteria for small probability. We will call this criteria significance level and denote it with α. α is in researcher’s control and usually set it to be 0.1, 0.01 or 0.05. If you calculate a probability for your sample which is less than α, you reject the null hypotesis. This is the essence of p-value approach Utku Suleymanoglu (UMich) Hypothesis Testing 12 / 37 Testing Hypothesis about the Population Mean: σ known Graphical recap: 0 Utku Suleymanoglu (UMich) Hypothesis Testing z 13 / 37 Testing Hypothesis about the Population Mean: 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. Compare the p-value with α. If p-value < α : Reject the H0 . You have enough evidence that H0 is false. If p-value > α : Fail to reject H0 . There is not enough evidence to reject the null hypothesis. 5. In our example, we reject the null if α is set to be 0.05 or 0.1 but not if 0.01. p-value approach allows easy comparison of decision with different α’s. 6. Notice: p-value is the smallest α choice where H0 is rejected. Utku Suleymanoglu (UMich) Hypothesis Testing 14 / 37 Testing Hypothesis about the Population Mean: σ known Critical Value Approach for One-Tailed Tests Another equivalently valid approach to create to criteria for testing would be this: 1. After hypotheses statement and the c...
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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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