handout6 - Estimation and Test Population parameters and...

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Estimation and Test Population parameters and Point Estimates Population Point Mean Variance Parameter Estimate [ E ( · )] [ V ar ( · )] μ ¯ X = n i =1 X i n μ σ 2 n σ 2 s 2 = n i =1 ( X i - ¯ X ) 2 n - 1 · · The sampling distribution of ¯ X is well approximated by a normal distri- bution with mean μ and variance σ 2 /n , i.e. , ¯ X N ( μ, ( σ/ n ) 2 ). Interval Estimation for population mean μ at (1 - α ) confidence level Population standard deviation σ in known : ¯ x ± z α/ 2 σ n Population standard deviation σ in unknown : ¯ x ± t α/ 2 σ n where t is with ( n - 1) degrees of freedom. Test for population mean(s) at α significance level Single Mean μ : H 0 : μ = μ 0 H 0 : μ μ 0 H 0 : μ μ 0 H a : μ = μ 0 H a : μ < μ 0 H a : μ > μ 0 σ known z = ¯ x - μ 0 σ/ n σ unknown t = ¯ x - μ 0 s/ n Reject | z | > z α/ 2 z < - z α z > z α | t | > t α/ 2 t < - t α t > t α where t is with ( n - 1) degrees of freedom. 1
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Two means μ 1 and μ 2 from independent populations: H 0 : μ 1 = μ 2 H 0 : μ 1 μ 2 H 0 : μ 1 μ 2 H a : μ 1 = μ
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