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Unformatted text preview: STA 2006 Hypotheses Testing: Asymptotics 20072008 Term II 1 General Theory: Known σ 2 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 : μ = μ vs. H 1 : μ 6 = μ The test statistics is defined as: T.S. = ¯ X μ σ / √ n where σ is the value of σ under H . For small n , the distribution of T.S. under H is complex and could even depend on μ . However, for large n , the distribution of T.S. is approximately N (0 , 1) under H by CLT. In that case, the decision rule is: Reject H 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 pvalue is defined as 2 × Pr { Z >  obs.T.S. } and the decision rule is equivalent to: Reject H if the pvalue 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 : 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 p V ar ( X ) /n = 31 . 7 25 p 25 / 100 = 13 . 4 For 5% test, z . 05 = 1 . 645 < obs.T.S. So, H is rejected at 5%. That is, the data give a strong evidence that λ is greater than 25. The pvalue is given by Pr { Z > 13 . 4 } < 10 6 . Example 2: Suppose the intertransaction 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 Summary for known σ 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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This note was uploaded on 06/05/2011 for the course STATISTICS 2006 taught by Professor Ho during the Spring '11 term at CUHK.
 Spring '11
 Ho
 Statistics, Variance

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