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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 < 2.5? Well we can
2.5 − 3
P (X < 2.5) = P (Z < ) = P (Z < z ) = P (Z < −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
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
1. After hypotheses statement and the calculation of test statistic (z for this case) :
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.
- Spring '08