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5102-Lecture-11 - Lecture 11 Stat 5102-004 11 February 2011...

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Lecture 11 Stat 5102-004 11 February 2011
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Confidence Regions Let B be a set whose elements are subsets of the parameter space ϑ . B ∈ B = B ϑ The region estimator ˆ B γ : Y → B is a level γ confidence region if Pr( θ ˆ B γ ) = γ for all θ ϑ . The confidence region is approximate if Pr( θ ˆ B γ ) γ for all θ ϑ . STAT 5102 (Theory of Statistics) Lecture 11 1 / 9
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Normal Mean, Variance Known Let Y 1 , . . . , Y n iid N( μ, σ 2 ) with σ 2 known. Pr ¯ Y - σ n z ( γ ) μ ¯ Y + σ n z ( γ ) = γ point estimate ± standard error × critical value ˆ B γ ( y ) = ¯ y + - σ n z ( γ ) , σ n z ( γ ) z (0 . 90) = 1 . 645 z (0 . 95) = 1 . 96 z (0 . 99) = 2 . 576 Pr( μ ˆ B γ ) = γ for all μ STAT 5102 (Theory of Statistics) Lecture 11 2 / 9
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Normal Mean, Variance Known, γ = 0 . 90 ● ● ●● ●● ●● ●● ●● ●● ● ● ● ●● ●●● ●● ●●● ● ● ●● ●● ●● ●● ● ● ●● ● ● ● ● ●● ●● μ - 2 σ n μ μ + 2 σ n μ - 2 σ n μ μ + 2 σ n STAT 5102 (Theory of Statistics)
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