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Dr. Hackney STA Solutions pg 180

Dr. Hackney STA Solutions pg 180 - 11-8 Solutions Manual...

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11-8 Solutions Manual for Statistical Inference Factor out the terms with y n and do some algebra on the middle term to get f ( y 1 , y 2 , . . . , y n ) = y Σ i λ i - 1 n e - y n 1 Γ( λ 1 ) 1 n - 1 j =1 (1 + y j ) λ 1 - 1 × n - 1 i =2 1 Γ( λ i ) y i - 1 1 + y i - 1 1 n - 1 j = i (1 + y j ) λ i - 1 × n - 1 i =1 1 (1 + y i ) n - 1 j = i (1 + y j ) . We see that Y n is independent of the other Y i (and has a gamma distribution), but there does not seem to be any other obvious conclusion to draw from this density. b. The Y i are related to the F distribution in the ANOVA. For example, as long as the sum of the λ i are integers, Y i = X i +1 i j =1 X j = 2 X i +1 2 i j =1 X j = χ 2 λ i +1 χ 2 i j =1 λ j F λ i +1 , i j =1 λ j . Note that the F density makes sense even if the λ i are not integers. 11.21 a. Grand mean ¯ y ·· = 188 . 54 15 = 12 . 57 Total sum of squares = 3 i =1 5 j =1 ( y ij - ¯ y ·· ) 2 = 1295 . 01 . Within SS = 3 1 5 1 ( y ij - ¯ y i · ) 2 = 5 1 ( y 1 j - 3 . 508)
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