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sta 2006 0708_chapter_8

sta 2006 0708_chapter_8 - STA 2006 Hypotheses...

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STA 2006 Hypotheses Testing (Section 8.2 and 8.3) 2007-2008 Term II 1 Simple Hypothesis vs Simple Hypothesis Suppose X 1 , . . . , X n are sampled i.i.d. N ( μ, σ 2 ) with known σ 2 . We know μ can only be either μ 0 or μ 1 . ( μ 1 > μ 0 ) . That is, H 0 : μ = μ 0 vs. H 1 : μ = μ 1 H 0 is called the null hypothesis . H 1 is called the alternative hypothesis . Task: Determine which hypothesis is more ”true” based on the sample information. A legitimate decision rule is to conclude H 1 is more true (reject H 0 ) if ¯ x n c where c is a real number. We are bounded to commit two types of error: H 0 is true H 1 is true Reject H 0 Type I error No Error Reject H 1 No Error Type II error 1
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2 1. Simple Hypothesis vs Simple Hypothe- sis The corresponding probabilities are: Let Z N (0 , 1). α = Pr( Committing Type I error ) = Pr μ = μ 0 ( ¯ X n c ) = Pr( Z c - μ 0 σ/ n ) = doubly crossed area in the figure (*) β = Pr( Committing Type II error ) = Pr μ = μ 1 ( ¯ X n < c ) = Pr( Z < c - μ 1 σ/ n ) = singly crossed area in the figure. We can decrease α by increasing c but then β will be increased. Similarly, β can be decreased by choosing a smaller c but then α will be increased. Conclusion: There is no way protecting both hypotheses with equal emphases. Statisticians decide to protect H 0 (”presumed innocence”) such that α is kept to be a small number (usually 0.05) and c can be fixed accord- ingly. 2
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How to fix c ? By ( * ) , Pr( Z c - μ 0 σ/ n ) = α c - μ 0 σ/ n = z α i.e. c = μ 0 + z α σ/ n . Therefore, the decision rule becomes: if ¯ X n μ 0 + z α σ/ n, reject H 0 . Interpretation: If such decision procedure is repeated many times, only 5% of the time H 0 will be wrongly rejected. A more formal way of writing the decision rule is: Reject H 0 if ¯ x n - μ 0 σ/ n z α . The random variable ¯ X n - μ 0 σ/ n is called the test statistics and ¯ x n - μ 0 σ/ n is called the observed test statistics (abbreviated as obs. T.S).
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