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sta 2006 0708_chapter_10

sta 2006 0708_chapter_10 - STA 2006 Hypotheses Testing...

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STA 2006 Hypotheses Testing: Asymptotics 2007-2008 Term II 1 General Theory: Known σ 2 0 Suppose X 1 , . . . , X n are i.i.d. sampled from a distribution (not necessarily normal) with mean μ and variance σ 2 and σ 2 is a function of μ . Consider the following hypotheses: H 0 : μ = μ 0 vs. H 1 : μ 6 = μ 0 The test statistics is defined as: T.S. = ¯ X - μ 0 σ 0 / n where σ 0 is the value of σ under H 0 . For small n , the distribution of T.S. under H 0 is complex and could even depend on μ 0 . However, for large n , the distribution of T.S. is approximately N (0 , 1) under H 0 by CLT. In that case, the decision rule is: Reject H 0 at level α if the absolute value of the observed test statistics is greater than or equal to the critical value z α/ 2 . The p-value is defined as 2 × Pr { Z > | obs.T.S. |} and the decision rule is equivalent to: Reject H 0 if the p-value is less than or equal to α . Example 1: Suppose the number of phone calls Sam received each day is i.i.d. Poisson with mean λ . For the last 100 days, his average number of phone calls per day is 31.7. Use α = 5% to test if λ is greater than 25. Solution: Assume n = 100 is large enough for CLT. Now the test is: H 0 : λ = 25 vs. H 1 : λ > 25 Note that V ar ( X ) = λ if X Poisson ( λ ). Therefore, the observed test statistics is obs.T.S. = ¯ x - λ 0 p V ar 0 ( X ) /n = 31 . 7 - 25 p 25 / 100 = 13 . 4 For 5% test, z 0 . 05 = 1 . 645 < obs.T.S. So, H 0 is rejected at 5%. That is, the data give a strong evidence that λ is greater than 25. The p-value is given by Pr { Z > 13 . 4 } < 10 - 6 . Example 2: Suppose the inter-transaction time of the shares of HSBC from 9:00 a.m. to 9:15 a.m. each trading day is exponentially distributed. A sample of 1000 observations is collected and the sample mean is 3.1 seconds. Test at 5% level that if the population mean is greater than 3 seconds. 1
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Summary for known σ 0 case : X 1 , . . . , X n are i.i.d. (not necessarily normal) with common mean μ and variance σ 2 which is a function of μ .
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