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Dr. Hackney STA Solutions pg 178

Dr. Hackney STA Solutions pg 178 - 11-6 Solutions Manual...

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11-6 Solutions Manual for Statistical Inference 11.14 Let X i n( θ i , σ 2 ). Then from Exercise 11.11 Cov i a i c i X i , i c i v i X i = σ 2 a i v i Var i a i c i X i = σ 2 a 2 i c i , Var (∑ i c i v i X i ) = σ 2 c i v 2 i , and the Cauchy-Schwarz inequality gives a i v i a 2 i /c i c i v 2 i . If a i = c i v i this is an equality, hence the LHS is maximized. The simultaneous statement is equivalent to k i =1 a i y i · - θ i ) 2 s 2 p k i =1 a 2 i /n M for all a 1 , . . . , a k , and the LHS is maximized by a i = n i y i · - θ i ). This produces the F statistic. 11.15 a. Since t 2 ν = F 1 , it follows from Exercise 5.19(b) that for k 2 P [( k - 1) F k - 1 a ] P ( t 2 ν a ) . So if a = t 2 ν,α/ 2 , the F probability is greater than α , and thus the α -level cutoff for the F must be greater than t 2 ν,α/ 2 . b. The only difference in the intervals is the cutoff point, so the Scheff´ e intervals are wider.
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