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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II Assignment 4 (Sketch Solution) Section A Hypotheses & likelihood ratios A1. Let X be the no. of marked fish among a sample of 2,000 caught from the lake and N be the total no. of fish in the lake. (a) (i) H : N ≤ 21000 vs H 1 : N > 21000. (ii) Note that the likelihood is ‘ ( N ) = ( 1000 100 )( N 1000 1900 ) / ( N 2000 ) . Consider ‘ ( N +1) /‘ ( N ) = 1 100( N 19999)( N +1) 1 ( N 2899) 1 > 1 , 2900 ≤ N ≤ 19998 , = 1 , N = 19999 , < 1 , N ≥ 20000 . Thus ‘ ( N ) increases as N increases up to 19,999, stays constant at N = 19999 and 20000, and then decreases as N increases beyond 20,000. Under H 1 , ‘ ( N ) is maximized at N = ˆ N 1 = 21000. Under H , ‘ ( N ) is maximized at N = ˆ N = 20000 (or 19999). The likelihood ratio is ‘ ( ˆ N 1 ) /‘ ( ˆ N ) = 0 . 872511. (b) (i) H : N > 21000 vs H 1 : N ≤ 21000. (ii) Under H , ‘ ( N ) is maximized at N = ˆ N = 21000. Under H 1 , ‘ ( N ) is maximized at N = ˆ N 1 = 20000 (or 19999). The likelihood ratio is ‘ ( ˆ N 1 ) /‘ ( ˆ N ) = 1 . 146118. A2. Let X s ,X m ,X n be nos. substantially, mildly and not improved by the ordinary treatment re spectively. Define Y s ,Y m ,Y n for the new treatment similarly. Then ( X s ,X m ,X n ) ∼ Multinomial(54; p s ,p m ,p n ) and ( Y s ,Y m ,Y n ) ∼ Multinomial(54; q s ,q m ,q n ) . (i) H : p s = q s ,p m = q m vs H 1 : p s < q s ,p m < q m . (ii) The likelihood is ‘ ( p,q ) ∝ p 12 s p 6 m p 36 n q 18 s q 12 m q 24 n . Under H 1 , ‘ is max’d at p s = 12 / 54, p m = 6 / 54, p n = 36 / 54, q s = 18 / 54, q m = 12 / 54 and q n = 24 / 54. Under H , ‘ is max’d at p s = q s = 30 / 108, p m = q m = 18 / 108 and p n = q n = 60 / 108. The likeihood ratio is 16 . 9724. A3. Let X be the no. of heads. Then X ∼ Binomial(100 ,p ), p ∈ [0 , 1]. (i) H : p = 1 / 2 vs H 1 : p 6 = 1 / 2. (ii) The likelihood is ‘ ( p ) ∝ p 60 (1 p ) 40 , maximized at p = ˆ p = 0 . 6. The likelihood ratio is ‘ (ˆ p ) /‘ (1 / 2) = 7 . 48987. 1 A4. (i) H : θ = ψ vs H 1 : θ < ψ . (ii) The likelihood is ‘ ( θ,ψ ) = θ 5 e 30 θ ψ 11 e 30 ψ . Under H , ‘ ( θ,θ ) is max’d at θ = ˆ θ = 4 / 15. Under H 1 , ‘ ( θ,ψ ) is max’d at θ = ˆ θ = 1 / 6 and ψ = ˆ ψ = 11 / 30. The likelihood ratio is ‘ ( ˆ θ, ˆ ψ ) /‘ ( ˆ θ , ˆ θ ) = 3 . 1676. A5. Let X 1 ,X 2 be the nos. of breakdowns on two consecutive days. Assume X i ∼ Poisson( λ ), i = 1 , 2, independently. (i) The probability in question is P ( X 1 = 0) = e λ . We want to test H : e λ > 3 / 4 (i.e. λ < ln(4 / 3)) vs H 1 : e λ ≤ 3 / 4 (i.e. λ ≥ ln(4 / 3))....
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This note was uploaded on 04/15/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.
 Spring '10
 SMSLee
 Statistics, Probability

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