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04_Testing_part1

# 04_Testing_part1 - Hypothesis testing What is hypothesis...

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NTHU MATH 2820, 2008, Lecture Notes Ch9, p.1 Hypothesis testing • What is hypothesis testing? Question 7.1 (What is a hypothesis testing question?) 1. observed data. x 1 , . . . , x n 2. statistical modeling. Regard x 1 , . . . , x n as a realization of random variables X 1 , . . . , X n , and assign X 1 , . . . , X n a joint distribution: a joint cdf F ( ·| Θ ) , or a joint pdf f ( ·| Θ ) , or a joint pmf p ( ·| Θ ) , where Θ = ( θ 1 , . . ., θ k ) , and θ 0 i s are fi xed constants , but their values are unknown . 3. point estimation. What is the value of Θ ? Find a function of X 1 , . . . , X n , Θ , to estimate Θ or a function of Θ . 4. hypothesis testing. Separate into two sets 0 and A , where 0 A = and 0 A = . Use the data X 1 , . . . , X n to answer the question: which of the two hypotheses, H 0 : Θ 0 versus H A : Θ A is more favorable. made by Shao-Wei Cheng (NTHU, Taiwan) Ch9, p.2 Definition 7.1 ( null and alternative hypotheses, simple and composite hypotheses, TBp.331,332,334 ) Question 7.3 (asymmetry between H 0 and H A ) Question 7.2 Is there a di ff erence between the roles of H 0 and H A ? Can we arbitrary exchange the two hypotheses? Can we obtain an estimate of Θ and accept H 0 if ˆ Θ 0 and reject H 0 if ˆ Θ A ? Hint . Consider the case: X Binomial( n, p ) , H 0 : p = 0 . 5 v.s. H 1 : p 6 = 0 . 5 , What if n = 10 5 and ˆ p = 0 . 50001? H 0 is called null hypothesis . H A (sometimes denoted as H 1 ) is called alternative hypothesis . An hypothesis is said to be a simple hypothesis if that hypothesis uniquely speci fi es the distribution.

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