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: 7.2......... 1 7.2.1....... 1 7.2.2......... 13 7.2.3................ 18 7.2.4 0-1 p.. 19 7.2.5.... 21 Previous Next First Last Back Forward 1
7.2 :, (0-1 ). 7.2.1, X N(µ, σ 2 ), < µ <, σ 2 > 0; X 1,, X n X. α. µ σ 2 : (1), H 0 : µ = µ 0 H 1 : µ µ 0. Previous Next First Last Back Forward 1
µ X, Z = Z(X 1,, X n ) = n X µ 0 σ H 0, U N(0, 1), Z, U z(x 1,, x n ), H 0. { Z > τ}. α, P H0 ( Z > τ) = α, τ = z α/2. { Z > u α/2 }. (x 1,, x n ) n x µ 0 σ > u α/2 Previous Next First Last Back Forward 2
H 0., H 0 : µ = µ 0 H 1 : µ > µ 0 H 0 : µ µ 0 H 1 : µ > µ 0 Z, Z H 0, {Z > u α }. H 0 : µ = µ 0 H 1 : µ < µ 0 H 0 : µ µ 0 H 1 : µ < µ 0 {Z < u α }. µ = µ 0 I α, µ µ 0 ( µ µ 0 ) I α. Previous Next First Last Back Forward 3
Z. 16, ( : ) : Example 2.942371 2.988662 3.106234 3.109316 3.118427 3.132254 3.140042 3.170188 2.902562 3.128003 3.146441 2.978240 3.103600 3.003394 3.044384 2.849916 0.1, α = 0.01, 3? α = 0.05? Example : µ, H 0 : µ = 3 H 1 : µ 3. Previous Next First Last Back Forward 4
Z = n( X 3)/0.1, Z > u α/2. z 2.16, 0.01, u 0.005 2.58,, 3 ; 0.05, u 0.025 = 1.96,, 3. :,. N(µ, σ 2 )( σ 2 ) H 0 : µ µ 0 H 1 : µ < µ 0 β > 0 Example Example β ϕ (µ) 1 β, µ < µ 0 (7.1) µ < µ 0 µ µ 0 β ϕ (µ) α α, β α < 1 β (7.1) Previous Next First Last Back Forward 5
µ 1 < µ 0 β ϕ (µ) 1 β, µ < µ 1 (7.2) β ϕ (µ) µ β ϕ (µ 1 ) 1 β ( ) n(µ0 µ) Φ u α 1 β σ n σ 2 (u α + u β ) 2 /(µ 0 µ) 2 Previous Next First Last Back Forward 6
(2) H 0 : µ = µ 0 µ µ 0,, X S 2 σ 2, T = n X µ 0. S H 0, T t n 1, { T > t n 1 (α/2)}. t., 7.2.1. Previous Next First Last Back Forward 7
( 7.2.1 ), 0.01 0.05 3? Example Example : µ, H 0 : µ = 3 H 1 : µ 3 T = n( X 3)/S, T > t n 1 (α/2). 2.21, 0.01 t 15 (0.005) 2.95,, 0.05 t 15 (0.025) 2.13,, 0.01 3, 0.05 3,. Previous Next First Last Back Forward 8
(3) H 0 : σ 2 = σ 2 0 H 1 : σ 2 σ 2 0., σ 2 χ 2 = 1 σ 2 0 ˆσ 2 = 1 n n i=1 n (X i µ) 2 i=1 (X i µ) 2 = nˆσ2 σ 2 0 H 0, χ 2 χ 2 n, χ 2 n, H 1, χ 2 = σ2 nˆσ 2 σ0 2 σ 2 σ2 n n, χ 2 n H σ0 2 0, { χ 2 < χ 2 n(1 α/2) χ 2 > χ 2 n(α/2) }.. Previous Next First Last Back Forward 9
, χ 2 = (n 1)S2, σ0 2 S 2. H 0, χ 2 χ 2 n 1, { χ 2 < χ 2 n 1(1 α/2) χ 2 > χ 2 n 1(α/2) }.,, 7.2.1. χ 2. Previous Next First Last Back Forward 10
7.2.1: N(µ,σ 2 ). µ (σ 2 Æ ) Z = Z > u α/2 n( X µ 0)/σ N(0,1) Z > u α Z < u α µ (σ 2 ) T = T > t n 1(α/2) n( X µ 0)/S t n 1 T > t n 1(α) T < t n 1(α) σ 2 (µæ ) χ 2 = 1 χ 2 > χ 2 n(α/2) χ 2 < χ 2 n(1 α/2) n σ0 2 i=1 (Xi µ)2 χ 2 n χ 2 > χ 2 n(α) χ 2 < χ 2 n(1 α) σ 2 (µ ) χ 2 = 1 χ 2 > χ 2 n 1(α/2) χ 2 < χ 2 n 1(1 α/2) n σ0 2 i=1 (Xi X) 2 χ 2 n 1 χ 2 > χ 2 n 1(α) χ 2 < χ 2 n 1(1 α) : µ µ 0, µ > µ 0 µ < µ 0. B : σ 2 σ 2 0, σ 2 > σ 2 0 σ 2 < σ 2 0. Previous Next First Last Back Forward 11
( 7.2.1 ) 0.1 0.1? Example Example : σ 2, H 0 : σ 2 0.1 2 H 1 : σ 2 > 0.1 2. χ 2 = (n 1)S2 0.1 2, {χ 2 > χ 2 n 1(α)}. χ 2 14.32, 0.2 χ 2 15(0.1) 22.31,, 0.1 0.1. Previous Next First Last Back Forward 12
7.2.2 X N(µ 1, σ 2 1), Y N(µ 2, σ 2 2), < µ 1, µ 2 <, σ 2 1, σ 2 2 > 0; X 1,, X n X, Y 1,, Y n Y.. µ 1 µ 2 σ 2 1/σ 2 2. α.,,, 10 9 ( : ), 62 57 65 60 63 58 57 60 60 58 50 59 56 57 58 57 56 55 57 N(µ 1, σ 2 ), N(µ 2, σ 2 ), µ 1, µ 2, σ 2. α = 0.1? Example Example Previous Next First Last Back Forward 13
: µ 1 > µ 2,, H 0 : µ 1 µ 2 = 0 H 1 : µ 1 > µ 2. µ 1 µ 2 X Ȳ T = X Ȳ. 1 S w + 1 m n H 0, T t m+n 2, {T > t m+n 2 (α))}. T t = x ȳ 1 s w + 1 m n = 3.23. Previous Next First Last Back Forward 14
α = 0.1 t m+n 2 (α/2) = t 17 (0.1) 1.33 < 3.23, H 0, 0.1. 7.2.2, σ 2 1 = σ 2 2 = σ 2., Example H 0 : σ2 1 σ 2 2 = 1 H 1 : σ2 1 σ 2 2 1. Example : σ 2 1 σ 2 2 ˆσ 1 2 = 1 m (X i m X) 2, ˆσ 2 2 = 1 n i=1 n (Y i Ȳ )2. i=1 Previous Next First Last Back Forward 15
θ = σ 2 1/σ 2 2 ˆθ = ˆσ 2 1/ˆσ 2 2 F = S2 1 S 2 2 = (m 1)ˆσ2 1/m (n 1)ˆσ 2 2 /n. F X Y., F F m 1,n 1. {F < F m 1,n 1(α/2) F > F m 1,n 1(1 α/2)}. F f = 1.19, α = 0.2, F 9,8 (0.1) = 2.44, F 9,8 (0.9) = 1/F 8,9 (0.1) = 0.41 ( X F m,n, 1/X F n,m ). 0.41 < 1.19 < 2.44, H 0, 0.2. Previous Next First Last Back Forward 16
7.2.2: Z > u(α/2) ( BÆ ) Z = X Ȳ N(0,1) Z > u(α) σ 1m 2 + σ2 2n Z < u(α) T > t m+n 2(α/2) ( B ) X Ȳ T = Sw 1 t m + 1 m+n 2 T > t m+n 2(α) n T < t m+n 2(α) m i=1 B( Æ ) F = (Xi µ1)2 /m n i=1 (Xi µ2)2 /n F m,n B( ) F = S2 1 S 2 2 F m 1,n 1 F > F m,n(α/2) F < 1 Fn,m(α/2) F > F m,n(α) F < 1 Fn,m(α) F > F m 1,n 1(α/2) F < 1 Fn 1,m 1(α/2) F > F m 1,n 1(α) 1 F < Fn 1,m 1(α) : µ 1 µ 2, µ 1 > µ 2 µ 1 < µ 2. B : σ 2 1 σ 2 2, σ 2 1 > σ 2 2 σ 2 1 < σ 2 2. B Previous Next First Last Back Forward 17
7.2.3,,. {(X 1, Y 1 ),, (X n, Y n )} X Y,, Z = Y X Z 1 = X 1 Y 1,, Z n = X n Y n, Z,! Previous Next First Last Back Forward 18
7.2.4 0-1 p,. (X 1,, X n ) X, 0-1, 1 p. : (1) H 0 : p = p 0 H 1 : p p 0 ; (2) H 0 : p = p 0 H 1 : p > p 0 H 0 : p p 0 H 1 : p > p 0 ; (3) H 0 : p = p 0 H 1 : p < p 0 H 0 : p p 0 H 1 : p < p 0. n, α. p X, T = X p 0 n p0 (1 p 0 ), p 0 p 0(1 p 0)/n X p = p 0, H 0, T N(0, 1). { T > u α/2, {T > u α } {T < u α } Previous Next First Last Back Forward 19
0.5.. 80, 5., 0.1,? Example Example : X B(1, p), p. α = 0.1, p = 0.05, H 0 : p = 0.05 H 1 : p 0.05. x = 5/80 = 0.0625, T t = x p 0 n p0 (1 p 0 ) = 0.513. α = 0.10 u 0.05 = 1.645., t < 1.645, Previous Next First Last Back Forward 20
H 0, 0.10. 7.2.5 X 1,, X n F (x; θ) θ 1 α [θ, θ], P (θ θ θ) 1 α H 0 : θ = θ 0 H 1 : θ θ 0 P (θ θ 0 θ) 1 α P (θ 0 > θ) + P (θ 0 < θ) α Previous Next First Last Back Forward 21
ϕ : θ θ 0 θ H 0,, H 0 : θ = θ 0 H 1 : θ θ 0 θ(x 1,, x n ) θ 0 θ(x 1,, x n ) P (θ θ 0 θ) 1 α θ 0 θ P (θ θ θ) 1 α θ 1 α θ H 0 : θ = θ 0 H 1 : θ θ 0 θ 1 α H 0 : θ = θ 0 H 1 : θ θ 0 Previous Next First Last Back Forward 22
θ 1 α 1 α (θ, ) ( (, θ)) α ( ) H 0 : θ θ 0 H 1 : θ > θ 0 ( H 0 : θ θ 0 H 1 : θ < θ 0 ) Previous Next First Last Back Forward 23