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3 7.3,, (X 1,, X n )., H 0 : X F Karl Pearson χ 2. : F ˆF n, D( ˆF n, F ), sup x R ˆF n (x) F (x). H 0, ˆF n F, D( ˆF n, F ). Karl Pearson F, Pearson. Previous Next First Last Back Forward 1

4 7.3.1 (1) X, X a 1... a k P p 1... p k p 1,, p k. n, a 1,, a k n 1,, n k., : H 0 : P (X = a 1 ) = p 1,, P (X = a k ) = p k.,.,, np 1,, np k, : Previous Next First Last Back Forward 2

5 a 1 a 2 a k np 1 np 2 np k n 1 n 2 n k,, n i /n p i, np i n i. Pearson T = k (n i np i ) 2 = (O E) 2. np i E i=1,, H 0, T k 1 χ 2. : T > χ 2 α(k 1) Previous Next First Last Back Forward 3

6 . 6,. : 600, 97, 104, 82, 110, 93, ? Example Example : 6, p 1,, p 6, H 0 : p i = 1/6, i = 1,, 6., 100, χ 2 (97 100) ( ) (82 100) ( ) (93 100) ( )2 =6.94, = 5 χ χ 2 5(0.2) 7.29,, 0.2. Previous Next First Last Back Forward 4

7 Example (Mendel) Mendel ( ) n = 8023 n 1 = 6022 n 2 = 2001 Mendel H 0 : π 1 = 0.75, π 2 = 0.25 Example : Mendel (H 0) µ 1 = nπ 1 = = , µ 2 = nπ 2 = = Pearson χ 2 Z= (O E) 2 =( ) 2 / ( ) 2 / =0.015 E df = 1 p value Mendel Fisher Mendel Previous Next First Last Back Forward 5

8 p value = Fisher (2) X, X a 1... a k P p 1... p k p i = p i (θ 1,..., θ r ), i = 1,..., k r θ 1,..., θ r. np i, p i ˆp i, χ 2 = k i=1 (n i nˆp i ) 2 nˆp i. Karl Pearson, χ 2 k 1 χ 2, R. A. Fisher k 1 r, k 1 r. Previous Next First Last Back Forward 6

9 100,. A a, 100 AA, Aa aa 30, 40, 30, 0.05 Hardy-Weinberg? Example Example : H 0 : Hardy-Weinberg. A p, Hardy- Weinberg 3 P (AA) = p 2, P (Aa) = 2p(1 p) P (aa) = (1 p) 2, H 0 : P (AA) = p 2, P (Aa) = 2p(1 p), P (aa) = (1 p) 2. H 0, ˆp 2, ˆp 2 (1 ˆp) 100 (1 ˆp) 2, ˆp 0.5, χ 2 Previous Next First Last Back Forward 7

10 , = 1 ( ) χ , 0.05 Hardy-Weinberg (1).. ( A) ( B).. ( A, : ) ( B, : ).,,, a, b, a b ( p268).,. Previous Next First Last Back Forward 8

11 H 0 : A B.. n, (i, j) n ij. p ij = P ( A, B i, j), (7.1) u i = P ( A i), (7.2) v i = P ( B j) (7.3) H 0 : p ij = u i v j i, j u i v j, a 1 + b 1 = a + b 2. û i = n i n, ˆv j = n j n. Previous Next First Last Back Forward 9

12 ( ). n i = b j=1 n ij, n j = a i=1 n ij. H 0, (i, j) nˆp ij = n i n j /n, H 0, a b i=1 j=1 (n ij nˆp ij ). χ 2 = a b i=1 j=1 (n ij n i n j /n) 2. (n i n j /n) χ 2 k 1 r = ab 1 (a + b 2) = (a 1)(b 1) χ 2., 1. (2), A B,. ; A,.,. Previous Next First Last Back Forward 10

13 .. Example 150(n 11 ) 88(n 12 ) 238(n 1 ) 36(n 21 ) 18(n 22 ) 54(n 2 ) 186(n 1 ) 106(n 2 ) 292(n) Example :. χ , 1 χ ,, Previous Next First Last Back Forward 11

14 7.3.3 (X 1,, X n) X, X F (x), r θ 1,, θ r. α H 0 : F (x) = F 0 (x; θ 1,, θ r ), F 0 (x; θ 1,, θ r )., H 0 : X N(µ, σ 2 ), r = 2, θ 1 = µ, θ 2 = σ 2. F 0 (x; µ, σ 2 ) = x { 1 exp 1 2πσ 2 2σ 2 (t µ)2 } dt.,. k (a j 1, a j ], j = 1,, k, Previous Next First Last Back Forward 12

15 a 0, a k., k. p j = P H0 (a j 1 < X a j ) = F 0 (a j ; θ 1,, θ r ) F 0 (a j 1 ; θ 1,, θ r ), j = 1,, k. H 0, p j (a j 1, a j ] f j = n j /n, n j. p i, χ 2 = k j=1 (n j np j ) 2 np j, χ 2 = k j=1 (n j nˆp j ) 2 nˆp j, ˆp i p i p i. Previous Next First Last Back Forward 13

16 { χ 2 > χ 2 k r 1(α) }. p i, r = 0. χ 2 n 50, nˆp j 5, j = 1,, k,,. 100, , : Example (, 1) [ 1, 0.5) [ 0.5, 0) [0, 0.5) [0.5, 1) [1, ) ? Example Previous Next First Last Back Forward 14

17 : µ σ 2, i (a i 1, a i, i = 1,, 6, i 100P (a i 1 < X a i ), X N(µ, σ 2 ). µ = σ 2 = = 1.622,. H 0 : χ , 6-1-2=3 χ χ 2 3(0.05) 7.81, Previous Next First Last Back Forward 15

18 P T (X) > τ, X 1 X 2, : T (X 1 ) > τ, T (X 2 ) > τ?? P = P ( H 0, T (X) T (x) ) P. α, ( ). Previous Next First Last Back Forward 16

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: p Previous Next First Last Back Forward 1 7-2: : 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 0;