sta 2006 0708_chapter_9

sta 2006 0708_chapter_9 - STA 2006 Hypotheses Testing:...

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Unformatted text preview: STA 2006 Hypotheses Testing: Variances (Section 8.2 and 8.3) 2007-2008 Term II 1 Test for σ 2 Suppose X 1 , .. . ,X n are i.i.d. sampled from N ( μ, σ 2 ). Both σ 2 and μ are unknown. The sample variance is S 2 = ∑ n i =1 ( X i- ¯ X ) 2 n- 1 . To test H : σ = σ vs. H 1 : σ > σ , note that ( n- 1) S 2 σ 2 ∼ χ 2 ( n- 1) under H . Therefore, the decision rule is: Reject H at level α if obs. T.S. = ( n- 1) s 2 σ 2 ≥ χ 2 α ( n- 1) Similarly, we can derive the decision rules of the one-tailed test for H 1 : σ < σ and the two-tailed test for H 1 : σ 6 = σ . In summary, H : σ = σ with obs. T.S. = ( n- 1) s 2 σ 2 and Y ∼ χ 2 ( n- 1) H 1 p-value Rej. H when σ 6 = σ 2 × Pr( Y > obs.T.S. ) if obs.T.S. > 1 2 × Pr( Y < obs.T.S. ) if obs.T.S. < 1 1. obs.T.S. ≥ χ 2 α/ 2 ( n- 1) or obs.T.S. ≤ χ 2 1- α/ 2 ( n- 1) 2. p-value ≤ α 3. σ 2 lies outside 100(1- α )% C.I. of σ 2 σ > σ Pr( Y > obs. T.S.) 1. obs. T.S. ≥ χ 2 α ( n- 1) 2. p-value ≤ α σ < σ Pr( Y < obs. T.S.) 1. obs. T.S. ≤ χ 2 1- α ( n- 1) 2. p-value ≤ α 1 Example 1: In a sample of credit card holders the mean monthly value of credit card purchases was $ 232 and the sample variance was 620. Assumeof credit card purchases was $ 232 and the sample variance was 620....
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sta 2006 0708_chapter_9 - STA 2006 Hypotheses Testing:...

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