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5102-Lecture-12

5102-Lecture-12 - Lecture 12 Stat 5102-004 14 February 2011...

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Unformatted text preview: Lecture 12 Stat 5102-004 14 February 2011 Confidence Intervals for Normal Random Sampling Y 1 , . . . , Y n iid ∼ N( μ, σ 2 ) S 2 = 1 n- 1 n X i =1 ( Y i- ¯ Y ) 2 Parameter Unknowns Sampling Distribution μ μ N(0 , 1) σ 2 σ 2 χ 2 ( n ) σ 2 μ, σ 2 χ 2 ( n- 1) μ μ, σ 2 t ( n- 1) ∑ n i =1 ( Y i- μ ) 2 σ 2 ∼ χ 2 ( n ) √ n ¯ Y- μ σ ∼ N(0 , 1) ∑ n i =1 ( Y i- ¯ Y ) 2 σ 2 ∼ χ 2 ( n- 1) √ n ¯ Y- μ S ∼ t ( n- 1) STAT 5102 (Theory of Statistics) Lecture 12 1 / 14 Student’s t with 1 Degree of Freedom x F(x)-6-4-2 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 t(1) N(0,1) x f(x)-6-4-2 2 4 6 0.0 0.1 0.2 0.3 0.4 STAT 5102 (Theory of Statistics) Lecture 12 2 / 14 Student’s t with 2 Degrees of Freedom x F(x)-6-4-2 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 t(2) N(0,1) t(1) x f(x)-6-4-2 2 4 6 0.0 0.1 0.2 0.3 0.4 STAT 5102 (Theory of Statistics) Lecture 12 3 / 14 Student’s t with 4 Degrees of Freedom x F(x)-6-4-2 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 t(4) N(0,1) t(1) x f(x)-6-4-2 2 4 6 0.0 0.1 0.2 0.3 0.4 STAT 5102 (Theory of Statistics) Lecture 12 4 / 14 Student’s...
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5102-Lecture-12 - Lecture 12 Stat 5102-004 14 February 2011...

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