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# 401hw8 - stat 401 Homework 8 solution(Spring 2008 7.2.4 1(a...

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stat 401 Homework 8 solution (Spring, 2008) 7.2.4 1. (a) θ = p ,the probbability the credit card customers who had in- curred an interest charge during the previous year because of an unpaid balance. (b) ˆ p = 136 / 200 = 0 . 68 (c) σ ˆ p = q p (1 - p ) n ˆ σ ˆ p = q ˆ p (1 - ˆ p ) n = q . 68(1 - . 68) 200 = 0 . 03298 (d) Note that y Bin ( n, p ) E p ) = Ey n = np n = p So the estimator is unbiased. 5. Note that X Bin ( m, p 1 ) , Y Bin ( n, p 2 ). (a) ˆ p 1 = x m , ˆ p 2 = y n . (b) E p 1 - ˆ p 2 ) = E ˆ p 1 - E ˆ p 2 = EX m - EY n = mp 1 m - np 2 n = p 1 - p 2 . (c) s.e. p 1 - ˆ p 2 ) = p V ar p 1 - ˆ p 2 ) = V ar ˆ p 1 + V ar ˆ p 2 = q p 1 (1 - p 1 ) m + p 2 (1 - p 2 ) n d s.e. p 1 - ˆ p 2 ) = p V ar p 1 - ˆ p 2 ) = V ar ˆ p 1 + V ar ˆ p 2 = q ˆ p 1 (1 - ˆ p 1 ) m + ˆ p 2 (1 - ˆ p 2 ) n = q X ( m - X ) m 3 + Y ( n - Y ) n 3 . (d) d p 1 - p 2 = 60 100 - 150 200 = - 3 20 . (e) d s.e. ( d p 1 - p 2 ) = q 60 × 40 100 3 + 150 × 50 200 3 = . 0577711. 6. X 1 , · · · , X 10 N ( μ, σ 2 ); Y 1 , · · · , Y 10 N ( μ, 4 σ 2 ). (a) E ˆ μ = δ E ¯ X + (1 - δ ) E ¯ Y = δ · μ + (1 - δ ) · μ = μ . (b) V ar ˆ μ = δ 2 V ar ¯ X + (1 - δ ) 2 V ar ¯ Y = δ 2 σ 2 10 + (1 - δ ) 2 4 σ 2 10 . (c) V ar ¯ X = σ 2 10 , V ar ˆ μ = σ 2 8 . I choose ¯ X because, its variance is smaller than ˆ μ ’s. 7. From the textbook, we have E ˆ α 1 = α 1 , E ˆ β 1 = β 1 . E ˆ μ Y | X = x = E α 1 + ˆ β 1 X | X = x ) = E ˆ α 1 + E ˆ β 1 x = α + β 1 x = μ Y | X = x . Therefore, ˆ μ Y | X = x is unbiased .

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401hw8 - stat 401 Homework 8 solution(Spring 2008 7.2.4 1(a...

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