2 2003 12 5 ( ) ( )
2 I 3 1 3 2 2? 6 3 11 4? 12 II 14 5 15 6 16 7 17 8 19 9 21 10 22 11 F 25 12 : 1 26
3 I 1 17 11 x 1, x 2,, x n x( ) x = 1 n n i=1 x i 12 (SD ) x 1, x 2,, x n s 2 s 2 = 1 n n (x i x) 2 s i=1 s = s 2 21 22 23 ( ) 2 :
4 1 : : 31 32 0, 1 n 41 µ ( ) x 42 σ 2 ( ) s 2 u 2 = 1 n 1 n (x i x) 2 i=1 51 4 1 (H 0 ) 2 3 4 52 5% 1% 61 62 3 -,,, 63
5 64 65 66 ( ) 2 : :
6 2 2? 2 2? 42 ( )? 2? 3 ( ) 21 x 1, x 2,, x n x = 1 n n x s 2 = 1 n i=1 n (x x) 2 i=1 u 2 = 1 n 1 n (x x) 2 i=1 2 s 2 ( s) u 2 ( u)
22 7 22 x 1, x 2,, x n ( ) ( ) µ ( ) σ 2 µ x µ x σ 2 s 2 σ 2? ( ) 1 σ 2 n (xi µ) 2 µ x 1 n (xi x) 2 1 n (xi µ) 2 c (xi c) 2 c x s 2 σ 2 u 2 = 1 n 1 n (x i x) 2 i=1 (23 ) σ 2 s 2
8 2 2? 23 u 2 n 1? 231 ( ) Y Y E(Y ) X (1, 2, 3, 4, 5, 6) ( 1/6) E(X) (1, 1, 1,, 2, 2, 2,, 6, 6, 6) 35 n x 1, x 2,, x n X n X E(X) n µ E(X) = µ (1) X µ X µ (X µ) 2 E[(X µ) 2 ] (X µ) 2
23 9 σ 2 E[(X µ) 2 ] = σ 2 (2) n X X 1 X E( X) µ E( X) = µ (3) X µ ( ) E( ) = X µ X µ E[( X µ) 2 ] E[( X µ) 2 ] = σ2 n (4) µ X σ 2 n
10 2 2? 232 n 1 u 2 (xi x) 2 = [(x i µ) + (µ x)] 2 = (x i µ) 2 + 2 (x i µ)(µ x) + ( x µ) 2 = (x i µ) 2 2( x µ) (x i µ) + n( x µ) 2 = (x i µ) 2 2( x µ)( x i nµ) + n( x µ) 2 = (x i µ) 2 2( x µ)(n x nµ) + n( x µ) 2 = (x i µ) 2 2n( x µ) 2 + n( x µ) 2 = (x i µ) 2 n( x µ) 2 1 (x i µ) 2 E[ (x i µ) 2 ] =E[(x 1 µ) 2 ] + E[(x 2 µ) 2 ] + + E[(x n µ) 2 ] =σ 2 + σ 2 + + σ 2 (2) =nσ 2 2 n( x µ) 2 E[n( x µ) 2 ] =n E[( x µ) 2 ] =n σ2 n =σ 2 (4) [ E (xi x) 2] = nσ 2 σ 2 = (n 1)σ 2 [ 1 ] E (xi x) 2 = σ 2 n 1 u 2 σ 2
11 3 2 s 2 = 1 n n (x i x) 2 i=1 u 2 = 1 n 1 n (x i x) 2 i=1 (x x) 2 ( ) (SS ) n 1 (df ) 3 X 1, X 2, X 3 X = X 1 + X 2 + X 3 3 SS = (X 1 X) 2 + (X 2 X) 2 + (X 3 X) 2 3 X 2 ( X = 10, X 1 = 9, X 2 = 10 X 3 11 ) 2 (MS ) 2
12 4? 4? 41 3 1 (, p30) 3 1 2 3 ( ) 3 2 3 (H 0 )? (H 0 : µ 1 = µ 2 )? (H 0 : µ 2 = µ 3 )? (H 0 : µ 1 = µ 3 ) (H 0 ) t ( ) 3,? (H 0 : µ 1 = µ 2 = µ 3 )
42 2 13 3 3 42 2 5% H 0 H 0 ( I ) 005 3 3 H 0 1 I, 1 095 3 = 014 10 040 A, B, C 3 A vs B, B vs C, C vs A 3 t A A vs B t A vs C t H 0 3 H 0 1 I 1 095 3 = 014, I I
14 II 1 (, p30) 3 1 2 3 3 5 8 7 4 4 6 6 3 8 3 3 9 3 7 10 7 9 9 6 8 8 5 7 9 3 3 7 5 4 8 47 56 81 61 10 10 10 A A 1 A 2 A 3 x 11 x 12 x 13 x 21 x 22 x 23 x 31 x 32 x 33 x n1 x n2 x n3 T 1 T2 T3 Ḡ n n n
15 5 1 ( ) 3 ( ) ( ) 1 1 (5) (µ 1 ) (ε 11 ) ( ) 1 (5) = (µ 1 ) + (ε 11 ) 2 (4) = (µ 1 ) + (ε 21 ) 1 (8) = (µ 2 ) + (ε 12 ) 1 (7) = (µ 3 ) + (ε 13 ) j(j = {1, 2, 3}) µ j j i X ij X ij = µ j + ε ij µ 1, µ 2, µ 3 µ µ 1 = µ + τ 1, µ 2 = µ + τ 2, µ 3 = µ + τ 3 µ τ 1, τ 2, τ 3 3 ( ) µ, j τ j j i X ij X ij = µ + τ j + ε ij
16 6 6 ε ij 0 {x 11, x 21,, x n1 } µ + τ 1 {x 12, x 22,, x n2 } µ + τ 2 {x 13, x 23,, x n3 } µ + τ 3 3 ( ) ( ) 1 3 3 135, 229, 114
17 7? H 0 : τ 1 = τ 2 = τ 3 τ 1, τ 2, τ 3? τ 1, τ 2, τ 3 3 X ij ( ) σt 2 otal 1 τ j ( {τ 1, τ 2, τ 3 } ) σa 2 ɛ ij σerror 2 X ij = µ + τ j + ɛ ij σt 2 otal σa 2 σerror 2 τ 1 = τ 2 = τ 3 = 0 σa 2 = 0 τ 1, τ 2, τ 3 σa 2 0 τ 1, τ 2, τ 3 0 σ 2 A? σa 2 µ Ḡ 1 ( ) σt 2 reat σ2 A = n P τ j 3 1
18 7 τ j ( T j Ḡ) ε ij (x ij T j ) x ij = Ḡ + ( T j Ḡ) + (x ij T j ) X ij = µ + τ j + ɛ ij x ij = Ḡ + ( T j Ḡ) + (x ij T j ) (a) x ij ( ) σt 2 otal (3 ) (b) ( T j Ḡ) σ2 A? (c) (x ij T j ) σerror 2? X ij = µ+ τ j +8 ε ij σt 2 otal σa 2 σerror 2 (a)?(b)?(c) MS T otal MS A MS Error x ij = Ḡ+ ( T j Ḡ) + (x ij T j ) (c) (b)
19 8 3 SS T otal = 3 SS A = j=1 i=1 j=1 i=1 SS Error = n (x ij Ḡ)2 n {( T j Ḡ) 0}2 = n 3 j=1 i=1 3 ( T j Ḡ)2 j=1 n {(x ij T j ) 0} 2 = 3 j=1 i=1 n (x ij T j ) 2 SS T otal = SS A + SS Error ( ) 1 Ḡ ( ) ( ) ( ) (x ij Ḡ) = ( T j Ḡ) + (x ij T j ) (5 61) = (47 61) + (5 47) (4 61) = (47 61) + (4 47) (8 61) = (56 61) + (8 56) (4 61) = (56 61) + (4 56) (7 61) = (81 61) + (7 81) (6 61) = (81 61) + (6 81) SS T otal = 14547 SS A = 6207 SS Error = 834 14547 6207 834
20 8 : x ij = Ḡ + ( T j Ḡ) + (x ij T i ) Ḡ x ij Ḡ = ( T j Ḡ) + (x ij T i ) 2 (x ij Ḡ)2 = ( T j Ḡ)2 + (x ij T i ) 2 + 2( T j Ḡ)(x ij T i ) (x ij Ḡ)2 = j i j n( T j Ḡ)2 + j (x ij T j ) 2 + i j 2( T j Ḡ)(x ij T j ) i 2( T j Ḡ)(x ij T i ) = 2 j i j = 2 j {( T j Ḡ) i {( T j Ḡ) 0} (x ij T i )} = 0 (x ij Ḡ)2 = j i j n( T j Ḡ)2 + j (x ij T j ) 2 i SS T otal = SS A + SS Error
21 9 (3 ) SS T otal =( 1) = 3n 1 SS A =( T j 1) = 3 1 SS Error = ( -1) = 3(n 1) (3n 1) = (3 1) + 3(n 1) ( ) MS T otal = SS T otal /(3n 1) MS A = SS A /(3 1) MS Error = SS Error /3(n 1) 1 : ( ) ( ) ( ) (x ij Ḡ) = ( T j Ḡ) + (x ij T j ) SS T otal = 14547 SS A = 6207 SS Error = 834 3n 1 = 29 3 1 = 2 3(n 1) = 27 MS T otal = 502 MS A = 3103 MS Error = 308
22 10 10 3 ( ) 101 σa 2 0 MS A σerror 2 MS A MS A MS Error F = MS A /MS Error 1 F = 3103/308 = 1004 F σa 2 0 1 σ2 A 0 ( ) 1
102 23 102 1 1 3 Ḡ T 1, T 2, T 3 MS T otal MS A MS Error 2 1 (σ A 0 ) e ij T j 1: 1 2:
24 10 MS A e ij MS Error F = MS A /MS Error MS A MS Error F 1 F 103 2 (23) 3 E(MS T otal ) = σ 2 T otal MS A E(MS A ) = σ 2 Error + nσ 2 A 2 MS A σ A 2 σ2 A σ2 Error MS Error E(MS Error ) = σ 2 Error 3 MS Error σerror 2 (H 0 ) MS A MS Error σerror 2 (H 0 ) MS A F = MS A /MS Error H 0 : σa 2 = 0 (σa 2 τ 1, τ 2, τ 3 ) 2 3
25 11 F F 51 4 1 (H 0 ) : H 0 : (τ 1 = τ 2 = τ 3 = 0, σa 2 = 0) 2 F 3 (6 ) F F (, ) F F F α% α% 1 : (2, 27) F 1% F > 549 4 F 1 : F = 1004 1% F F (σt 2 otal )
26 12 : 1 12 : 1 6 1 2 3 4 F 5 1-5 1 1 (k, ) x ij i (1 n k ) j (1 k) ( ) A A 1 A 2 A k x 11 x 12 x 1k x 21 x 22 x 2k x 31 x 32 x 3k x n1 1 x n2 2 x nk k T 1 T2 Tk Ḡ n 1 n 2 n k 1 ( ) (SS) (df) (MS) F (A) k n j ( T SS A MS A j Ḡ)2 k 1 k 1 j=1 (Error) n k j (x ij T k j ) 2 SS Error (n j 1) k j=1 (n j 1) j=1 i=1 j=1 n k j k (x ij Ḡ)2 n j 1 j=1 i=1 j=1 MS Error