sta 2006 0708_chapter_8

sta 2006 0708_chapter_8 - STA 2006 Hypotheses Testing...

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Unformatted text preview: 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 or μ 1 . ( μ 1 > μ ) . That is, H : μ = μ vs. H 1 : μ = μ 1 • H 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 ) if ¯ x n ≥ c where c is a real number. We are bounded to commit two types of error: H is true H 1 is true Reject H Type I error No Error Reject H 1 No Error Type II error 1 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 ( ¯ X n ≥ c ) = Pr( Z ≥ c- μ 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 (”presumed innocence”) such that α is kept to be a small number (usually 0.05) and c can be fixed accord- ingly. 2 • How to fix c ? By ( * ) , Pr( Z ≥ c- μ σ/ √ n ) = α ⇒ c- μ σ/ √ n = z α i.e. c = μ + z α σ/ √ n . Therefore, the decision rule becomes: if ¯ X n ≥ μ + z α σ/ √ n, reject H . • Interpretation: If such decision procedure is repeated many times, only 5% of the time H will be wrongly rejected. • A more formal way of writing the decision rule is: Reject H if ¯ x n- μ σ/ √ n ≥ z α ....
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sta 2006 0708_chapter_8 - STA 2006 Hypotheses Testing...

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