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Dr. Hackney STA Solutions pg 148

# Dr. Hackney STA Solutions pg 148 - 9-6 Solutions Manual for...

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9-6 Solutions Manual for Statistical Inference 9.15 Fieller’s Theorem says that a 1 - α confidence set for θ = μ Y X is θ : ¯ x 2 - t 2 n - 1 ,α/ 2 n - 1 s 2 X θ 2 - 2 ¯ x ¯ y - t 2 n - 1 ,α/ 2 n - 1 s Y X θ + ¯ y 2 - t 2 n - 1 ,α/ 2 n - 1 s 2 Y 0 . a. Define a = ¯ x 2 - ts 2 X , b = ¯ x ¯ y - ts Y X , c = ¯ y 2 - ts 2 Y , where t = t 2 n - 1 ,α/ 2 n - 1 . Then the parabola opens upward if a > 0. Furthermore, if a > 0, then there always exists at least one real root. This follows from the fact that at θ = ¯ y/ ¯ x , the value of the function is negative. For ¯ θ = ¯ y/ ¯ x we have ( ¯ x 2 - ts 2 X ) ¯ y ¯ x 2 - 2 (¯ x ¯ y - ts XY ) ¯ y ¯ x + ( ¯ y 2 - as 2 Y ) = - t ¯ y 2 ¯ x 2 s 2 X - 2 ¯ y ¯ x s XY + s 2 Y = - t n i =1 ¯ y 2 ¯ x 2 ( x i - ¯ x ) 2 - 2 ¯ y ¯ x ( x i - ¯ x )( y i - ¯ y ) + ( y i - ¯ y ) 2 = - t n i =1 ¯ y ¯ x ( x i - ¯ x ) - ( y i - ¯ y ) 2 which is negative. b. The parabola opens downward if a < 0, that is, if ¯ x 2 < ts 2 X . This will happen if the test of H 0 : μ X = 0 accepts H 0 at level α .
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