Dr. Hackney STA Solutions pg 149

Dr Hackney STA - Second Edition 9-7 x interval.000.305 1.305.634 2.362.762 3.695,1 9.19 For F T t | θ increasing in θ there are unique values θ

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Unformatted text preview: Second Edition 9-7 x interval [.000,.305) 1 [.305,.634) 2 [.362,.762) 3 [.695,1]. 9.19 For F T ( t | θ ) increasing in θ , there are unique values θ U ( t ) and θ L ( t ) such that F T ( t | θ ) < 1- α 2 if and only if θ < θ U ( t ) and F T ( t | θ ) > α 2 if and only if θ > θ L ( t ). Hence, P ( θ L ( T ) ≤ θ ≤ θ U ( T )) = P ( θ ≤ θ U ( T ))- P ( θ ≤ θ L ( T )) = P F T ( T ) ≤ 1- α 2- P F T ( T ) ≤ α 2 = 1- α. 9.21 To construct a 1- α confidence interval for p of the form { p : ` ≤ p ≤ u } with P ( ` ≤ p ≤ u ) = 1- α , we use the method of Theorem 9.2.12. We must solve for ` and u in the equations (1) α 2 = x X k =0 n k u k (1- u ) n- k and (2) α 2 = n X k = x n k ` k (1- ` ) n- k . In equation (1) α/ 2 = P ( K ≤ x ) = P ( Y ≤ 1- u ), where Y ∼ beta( n- x,x + 1) and K ∼ binomial( n,u ). This is Exercise 2.40. Let Z ∼ F 2( n- x ) , 2( x +1) and c = ( n- x ) / ( x + 1). By Theorem 5.3.8c, cZ/ (1 + cZ ) ∼ beta( n- x,x + 1) ∼ Y...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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