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( ) X M(X ) . , n x xi (1.1), - i 1 D(X ) , 2 (1.7) X s (1.6). , . . , (1.8)), M(X ) ( D - D(X ) (1.7). : M ( X C) M ( X ) C , D( X C) D( X ) , C, x. 1.4. . f ( x) 1 e 2 9 ( x a )2 2 2 , (1.14) a – , ( 2 : M ( X ) a ; D( X ) (1.14) ) . x. x a x a; x a . x - . f (x ) . 1.1. . 1.1 : 1) a 2, - 1; 2) a 1, 1; 3) a 2, 2. f(x) 0,5 0,4 1 2 3 0,3 0,2 0,1 x 0 -1 0 1 2 3 4 5 . 1.1. a , – , , f (x) . , . , , , 1. X ( , P( ) X a ) (t ) – a , (1.15) : t (t ) ( z )dz , ( z) 0 (z ) (z ) ( (t ) . 1), 1 e 2 z2 2 . (1.16) 1 10 2. - , (1.14) , a 0 1– , , (t ) – z. , t , 1/2. X - d P( X , a d d) 2 . (1.17) , P( X a 3 ) 2 (3) 0.9973. , 0.0027 . , ( ). - ? , : , , - , , , . , , . , a (1.14), x: , f ( x) , f( x x) a , . . x 1 e 2 ( x x )2 2 2 , x x (1.18) , g ( x) , - g( x) 1 e 2 11 ( x )2 2 2 . (1.19) 1.5. , - ( d, d) x P( x (t ) – d ) 2 (d ), (1.20) (1.16). a: x, x Y x a. x/ Y , - 1. . (1.20) P x P( x t 2 (t ) t ) 2 (t ) , – ). ( , t x t , (1.21) t / 2. , 1.2. , 1 0.95 . 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( , , (1.21) ). , H 1 : a1 . a2 H 1 : a1 a2 , , ( z cr ) z cr , ( zcr ) 2 ( x1 ) 2 1 2 . - ( x2 ) , . s 2 ( x1 ) , 2 ( x1 ) s 2 ( x2 ) . , s 2 ( x1 ) , 2 ( x2 ) s 2 ( x2 ) . 2.2. . , : x1 , 24 36 , x 2 100 38 . - s 2 ( x1 ) 12 s 2 ( x2 ) 13 . , - . 0.02 . ? H 1 : a1 a2 36 38 12 / 100 13 / 100 Z 0.49 ; zcr ( z cr ) - H 0 : a1 a2 = 0.02. 2.33 ; Z 4; z cr – . - . 2.3. , , ), ( . . . . . 2.3. : 3, 4, 6, 10, 13, 17 ; x2 x1 1, 2, 5, 7, 16, 20, 22 . , n1 , n2 . : ( ) 1 2 3 4 5 6 7 10 13 16 17 20 22 1 2 3 4 5 6 7 8 9 10 11 12 13 , . – ( ) - : n1 W ri ; i 1 W 0.01. 3 4 6 8 9 11 41. . 6. , H 1 : a1 a2 , 25 (2.5) - . wcr ( W : wcr ( .) , .) n1 24 , wcr ( .) 7. 6, n2 60 . - wcr ( .) W . . , . W H 1 : a1 wcr ( .) 25 , wcr ( H 1 : a1 a 2 , , wcr ( .) 59 . - .) . wcr ( W a2 , . .) . , , - . 1. , , . 2. n1 n2 , - . . , ( . . 6) n1 10, n2 10 . W W ( ) , : M (W ) n1 (n1 n2 1) ; 2 (2.6) (W ) n1n2 ( n1 n2 1) . 12 (2.7) , Z W M (W ) (W ) (2.8) . H 0 : a1 (2.5), : 26 , a2 - W M (W ) H 1 : a1 W zcr H0, H1 . (2.9) a2 ( zcr ) )/2. (1 H 1 : a1 . a2 a2 , H 1 : a1 z cr , ( zcr ) ( z cr ) 1 2 . 2.4. , x2 , : 10 x1 22, 34, 52, 62, 30, 40, 64, 84, 56, 59 ; 52, 71, 76, 54, 67, 83, 66, 90, 77, 84 . 0.05 , . : - ( ): 22 30 34 40 52 52 54 56 59 62 64 66 67 71 76 77 83 84 84 90 1 2 3 4 5.5 5,5 7 8 9 10 11 12 13 14 15 16 17 18.5 18,5 20 , . – ( - ) : W 1 2 3 4 5.5 8 9 10 11 18.5; W 72. . 6. 0.05 , wcr ( .) 82 , wcr ( H 1 : a1 , .) 128 . W a2 , wcr ( .) . , , , . (2.6) M (W ) , (2.7) 10 21 105 , (W ) 2 ( zcr ) , 27 10 10 21 13.23. 12 1 0.05 0.45 , z cr 2 1.64 . . - W M (W ) W 33; M (W ) W zcr W zcr 21.7 ; H1 . , , x1 . 2.2. 50.3 x2 , s 2 ( x1 ) 315.6 , 2 72 2 : ( x2 ) s 2 ( x2 ) 141.6 . Z Z (2.3): 50 .3 72 315 .6 141 .6 10 10 ( zcr ) , Z ( x1 ) 3.2 . 1 2 0.05 0.45 , zcr 1.64. H1 . zcr , : , - . 2.4. 2 ). , , - . x1i , i 1, n1 , , x2 j , j 1, n2 . , u- , n1u n 2 u , u 1, k , k – . : 1 2 … u … l 1 n11 n12 … n1u … n1l n1 2 n21 n22 … n2u … n2 l n2 28 m1 m2 … mu … ml n 2 k 2 u nu , . (nu nu ) 2 , n 1 u u- (2.10) , , , nu - . - ( ) 2 k n1n2 u 2 cr ( k 1 1 1 n1u n2 n n2 2 . (2.11) , - , k 1) 2 . 5). n2 u n1u n1 2 cr ( ( , k 1) , – ). , , , n 50 . 2.5. , . ( 6 38 10 7 62 10 7 2 ) , : 1 1 9 13 17 8 2 50 1 (38–42) 2 (42–46) 3 (46–50) 4 (50–54) 5 (54–58) 6 (58–62) 2 2 7 18 15 6 2 50 3 16 31 32 14 4 100 , 0.05 . 2 (2.11) 2 cr ( 0.05, 5) 11.07 . 5. 1.8 . - . . 29 : 1. n1 8 10 : n2 19.68, 15.17, 20.54, 10.12, 22.69, 18.07, 22.65, 18.27 ; 23.22, 23.04, 22.44, 20.64, 17.67, 19.47, 19.27, 15.29, 16.4, 17.45 . x1 x2 ( 0.05 ) , : . . x1 1. 2 ( x1 ) 4 x2 , 2 ( x2 ) 3 , - Z (2.3) c z cr . . 2. 2 2 , ( x1 ) , ( x2 ) 2 s ( x1 ) 3. - 2 , s ( x2 ) . 1. , ( ( - . 6). ). 4. , , W (2.6), (2.7) - . 1 x1 1. x2 Excel , ABS. z cr ( zcr ) (1 ( . 2) . )/2, s 2 ( x1 ) 2. s 2 ( x2 ) - Excel . 3. : – (1 2), ( . – « ») ( - 30 ). n1 n 2 1 – , 18). ) ( , ), (1- , – , - 1( ), – (3- ). M (W ) (2.6) 4. (2.9). (W ) (2.7) , z cr . 1. ? 2. 2 ( n1 50 55 ): n2 x2 : x1 : 28.3 35.1 28.0 25.9 26.4 27.8 27.0 33.1 32.0 29.9 27.4 33.9 31.6 29.0 30.1 29.3 32.9 29.5 30.6 32.8 31.7 27.9 26.7 25.9 33.8 31.5 34.8 34.1 24.1 33.4 24.4 29.9 30.4 31.6 33.4 27.0 27.6 29.1 29.0 34.7 33.8 28.6 32.7 30.5 28.2 30.0 26.9 31.1 24.0 24.8 25.8 25 28.3 34.0 35.1 25.8 24.5 26.8 29.8 27.9 28.9 32.1 28.7 31.9 29.4 31.8 28.5 35.9 30.6 25.5 29.6 32.5 28.0 28.4 32.7 29.3 32.9 29.3 24.5 30.9 29.6 28.5 28.2 30.5 30.4 24.3 32.6 30.6 30.6 27.1 26.7 30.4 29.4 30.3 31.8 25.9 32.4 28.4 31.4 34.5 28.2 29.3 29.0 31.2 29.0 0.05 - . 2 2 . - k n k 1 3.332lg n. 50 7 . 24.0 35.9, 60, , R x max 7, ( - x min 1.7. , : 24.0 25.7, 25.7 27.4, 27.4 29.1, 29.1 30.8, 30.8 32.5, 32.5 34.2, 34.2 35.9. . : 31 ) ( ). , . . F2 ), CTRL+SHIFT+ENTER. , . 2 2 cr ( - , k 1) , . . , . 2.2. 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 1 2 3 4 5 6 7 . 2.2. 1. ? 2. 3. ? ? 4. 5. 6. 7. . ? ? ? 32 3. . . . - . 3.1. ( 2.1). , - , ( ) 0 . H0 : 2 : H1 : 2 0 ( x1 ) . H1 : 2 2 2 0, ( x1 ) x1 2 0. ( x1 ) - , , – 2 , . (n 1) s 2 ( x1 ) 2 0 H1 : 2 . (3.1) 2 0 ( x1 ) – 2 cr ( 2 , n 1) . ). . 2 cr , - H0 - H0 . H1 : 2 2 0 ( x1 ) 2 cr (1 – 2 , n 1) . ). 2 cr . , ( H0 - H0 . 3.1. 17 s 2 ( x1 ) 0.24 , 2 0 , 0.18 . 0.05 H0 : 2 ( x1) 2 0, H1 : , (3.1) ). 33 2 ( x1 ) 2 0. 16 0.24 21.33 . 0.18 2 ( H0 . 2 cr . 4) (0.05; 16) 2 26.5 . . . 2 cr , - . 3.2. , , 2 , , , , . ( ) ( , ). , ( ) - . . , - , , ( ) , ( . , , ) , - ( ). . : 2 1 H0 : 2 2. x1 x2 n1 - n2 s 2 ( x1 ) . s 2 ( x2 ) . , 2 2 s ( x1 ) > s ( x2 ) . F s 2 ( x1 ) s 2 ( x2 ) . (3.2) , k2 F, k1 n1 n2 1 . ( . 7), k1 k 2 . H1 : 34 2 1 2 2 , H1 : 2 1 2 2 . 1 P( F Fcr ) . Fcr ( , k1 , k 2 ) . , - P( F Fcr ) , Fcr ( 2 2, , k1, k2 ) . , , , , . , H0 : 2 1 2 2 . H1 : 2 1 2 2 . Fcr ( , k1 , k 2 ) ; 2 1 H1 : - F Fcr H0 . 2 2 Fcr ( 2 , k1, k2 ) . F ; - Fcr H0 . 3.2. . 2= 0.01 : 8.1, 7.8, 8.0, 8.2, 8.1, 8.1, 7.9, 7.8. 8.3, 8.5, 8.0. , 2, 1- ? , , , , . 1: 7.9, 7.7, : 2 s ( x2 ) 0.023, : s 2 ( x1 ) 0.102 , Fcr (0.01, 4, 7) 7.85 . F 4.4. F Fcr , . : , ! .( . , , ). , , - . « 0.05 Fcr ( 0.05, 4, 7) 4.12 , , , . 35 ». ( n 12 ), 3.9 . 3.3. 2 2 s ( x2 ) 5.7, s ( x1 ) , 0.02 2 1 H0 : 2 2 H1 : 2 1 2 2 . , Fcr ( 2 , k1, k2 ) Fcr (0.01, 11, 11) 4.46, 1.46. F Fcr , . - . 3.3. . , - , - . . . m . , c - . , , ( , !) , , , , , . . , . 2.2. . . , , , , m , . , ( C52 ) !) 10 - . , - . . , , ). ( . 36 k 1, 2, ..., m – « ( - »); nk – m k; n nk – ; x ki – , k 1 i k . - k : 1 nk xk nk x ki . (3.3) i 1 : 1 m nk x ki n k 1i 1 x 1 m nk x k . nk 1 (3.4) . SS Sumof Squares ). m nk ( xki SST x )2 , 2 1 SST n ( x) k 1i 1 ( , . ) m SSA (3.5) ( xk x ) 2 nk , , 2 A k 1 1 SSA n (3.6) - . , , . . - . . m nk ( xki SSR x k )2 , 2 R k 1i 1 1 SSR n (3.7) « . , ( ) . . ( ) 37 : » 2 1 nk ( xk ) nk x k )2 ; ( xki 2 R i 1 ( 1 m nk 1 ) 2 ( xk )nk . (3.8) : 2 A 1 m 2 ( x k nk nk 1 2 R m k 1 2x n 2 x k xnk x nk ) nk m 2 x k nk 2 x 1 m nk 2 ( xki n k 1i 1 2 2 2 x2 xk k 1 m xki i 1 2 xki x k 2 x k nk x2 k 1 n 2 xk ) n x 2 2 . , 2 . 2 A 2 R, SST SSA SSR . ( (3.9) ) . , . , - , . 2 0 SSA 1 SST (3.10) , - . 3.4. . 3.1 20 ( ), ( 4 - 5 ). . 3.1 1 ) 1 2 3 4 2 » 1.8 2.1 1.8 1.9 3 » » 2.1 2.3 2.4 2.6 . 3.2. 38 4 1.8 1.6 1.9 2.1 5 » 1.6 1.8 1.3 1.5 » 1.9 2.4 2.2 2.3 3.2 xk 2 ( xk ) 1 1.9 2 2.35 3 1.85 (k) 4 1.55 5 2.2 0.015 0.0325 0.0325 0.0325 0.035 2 x = 1.97. 2 2 R 2 A 2 R 2 A 0.1081 , 0.0295 . : 0.0786 , - 2 , 0.727 . 72.7 % , 27.3 % . . , . - , . . , 3.4 2.33 ( Fcr ( 0.05, 3, 3) ). 9.28 , . 3.4. . . , 3.4, , . « , » « », « 1.5 , , . » . , , - , . ( – ) . . (3.2). . . n 1 (3.5) (Mean Square) MST SST n 1 39 - , s2 ( x) n n 1 2 ( x) . (3.11) m 1 (3.6) , , - : MSA SSA m 1 2 A. m 1 (3.7) , , n s 2A (3.12) n m , : MSR SSR n m n s R2 « » 2 R. n m (3.13) , « , » : n 1 m 1 (n m) . (3.14) - A , : MSA . MSR F F (3.15) Fcr ( , m 1, n , . Fcr A 3.4 ( m) . , ). 20 20 5 20 2 2 0.393 , sR A 5 1 Fcr ( 0.05, 4, 15) 3.06 . F Fcr , s 2A 2 R 0.03933, F s 2A s R2 9.99 , . . 3.5. x B. , , . . Ak ( k 1, m ), Bi ( i 1, l ), N, . – n mlN 40 - , A n B. ml . - : A … … … … … … A1 x11 … x1i … x1l B1 … Bi … Bl … Ak xk1 … x ki … x kl … … … … … Am x m1 … xmi … x ml , . 1. ( Ak Bi ). n xki , ml . : xk 1 l xki , li 1 x xi 1 m xki . mk 1 (3.16) 1 m l xki . n k 1i 1 (3.17) : – ( , m l ) ( xki x )2 ; (3.18) ( xk x )2 ; (3.19) x)2 ; (3.20) SST k 1i 1 – A m SSA l k 1 – B l SSB m ( xi i 1 – ( 41 ) m l SSR ( xki xk x )2 . xi (3.21) k 1i 1 SST SSA SSB SSR , (3.22) ml 1 (m 1) (l 1) (m 1)(l 1) . (3.23) - : MSA – – 1 m 1 SSA ; MSB A B 1 l 1 SSB ; MSR (3.24) : MSA MSR Fcr ( , m 1, ( m 1)(l 1)) ; MSB MSR Fcr ( , l 1, (m 1)( l 1)) . (« Ak Bi , , 2. » 1 SSR . (m 1)(l 1) ). x (ki1) , x (ki2 ) , … x (kiN ) , N n mlN . : x ki 1 N N xki( j ) . (3.25) j 1 : xk 1 l x ki , li 1 x xi 1 m x ki . mk 1 (3.26) 1 m l x ki . n k 1i 1 (3.27) : – ( , N m l SST j 1 k 1i 1 – ( x (kij ) ) x) 2 ; (3.28) A m SSA lN ( xk k 1 42 x)2 ; (3.29) – B l SSB mN x )2 ; ( xi (3.30) i 1 – m A B xk xi l SSAB N ( x ki x )2 , (3.31) k 1i 1 – ( N m l ) ( xki( j ) x ki )2 . SSR (3.32) j 1k 1i 1 SST SSA SSB SSAB SSR , (3.33) : Nml 1 (m 1) (l 1) (m 1)(l 1) ml( N 1) . (3.34) - : MSA 1 m 1 SSA ; MSB 1 l 1 SSB ; (3.35) MSAB 1 SSAB ; MSR (m 1)(l 1) 1 SSR . ml ( N 1) : – A MSA MSR – B , Fcr ( , m 1, ml ( N 1)) ; , MSB MSR Fcr ( , l 1, ml ( N 1)) ; – MSAB MSR , Fcr ( , ( m 1)( l 1), ml ( N 1)) . - . 43 : 1. , , - . , . - . , , , . , , . . – A1 , A2 , A3 , A4 . , ( . ), : A1 A2 A3 A4 18.5 24.0 17.2 19.9 18.0 26.3 25.3 24.0 21.2 24.5 20.6 25.2 20.8 24.7 22.9 25.4 19.9 22.6 17.5 20.4 1 Excel . , , . 1. (3.3) (3.4). . 2. SST , , . 2 ( x) ( . - ) n 20 . 3. nk SSA (3.6). 5 , xk . 44 4. SSR (3.7) ( , - SST n 1 19 , SSA - ), . 5. : m 1 3, SSR n m 16 . MST, MSA, MSR (3.11–3.13). MSA . MSR F 6. 7. Fcr ( 0.05, 3, 16) .7 8. F . , « » - , (3.10). . , . , - . 3.1). . 3.1. ( 45 ) , . 3.2. . 3.2. ( . « » ) , . 3.2 – , ( Excel « » ). , . 1–8. , (3.46 3.23) , « . » (3.10) 2 SSA 63.2855 0.394 SST 160.7895 , 39.4 %. , 60.6 % . 46 2. . . , , . , 1- , , , , ? , ? , , , . 1, 21- , , , : A1 A2 A3 A4 20.6 18.0 19.0 21.3 13.2 22.6 24.6 19.6 23.8 27.1 27.7 18.6 20.8 25.1 17.7 21.5 20.0 21.1 23.9 16.0 2 , « 2 ». 4 B – 5 , m 4, l . A ). 2 , N 5, n 40 . , , 1: 1. x ki (3.25) . 2. 3. 4. 5. (3.26) (3.27). SST (3.28). SSA (3.19) SSB(3.20). SSAB (3.31). 1, . SSR (3.32) 6. , , , 5 , . 47 SST n 1 39 , SSA SSAB (m 1)(l 1) 3 , SSR ml( N 1) 32 . 7. : m 1 3, SSB l 1 1 , MST, MSA, MSB, MSAB, MSR (3.35). FA 8. MSA , FB MSR MSB , FAB MSR MSAB MSR FAcr ( 0.05, 3, 32) FBcr ( 0.05, 1, 32) FABcr ( 0.05, 3, 32 ) . , . , . , - ( ) ( . 3.3). , . 3.4. . ( , ( ) ). , . , - , – ) ( ( ). . 3.3. ( 48 ) . 3.4. ( ) (0.05), : , . FAB , . 3.4, F = 0.01 < 2.90, . 3.4, F) , FAcr F. , . . 49 , ). , - A , . FA , 5.20 2.90 , FAcr - , , FB B– , . 0.81 FBcr , , 4.15 . , , . 1. . 2. , - ? 3. ? 4. 5. 6. ? . - ( ). 4. . . . . - . 4.1. . ( , ) . , , , , v0 , H v02 / 2 g , , g = 9.81 /c2 – . , , . , . , . - , , . , . . - 50 . , v0 ( - , ) H. x y , x y, y - , f (x) . , , , . X Y, . 4.1. . , , ( , , .), - . , . , . . ( - , ) - . . ). - . . y j , j 1, m . - , , , X Y . xi , i 1, k , : pij , p( xi , y j ) . P( X xi ) pij ; P(Y j yj) pij i 51 P( X xi Y yj) pij / P(Y y j ); P(Y yj X xi ) pij / P( X xi ). – , ( xi ): (Y) (X) M (Y X xi ) y j P (Y xi ) . yj X (4.1) j M (Y X x) (x) ( x) , Y X. . , .( ! - , X Y). ( ) , , , . . , . « » - , X Y. , . . ) X Y : ( X ,Y ) M [ X M ( X )][Y M (Y )] ( X ,Y ) M ( XY ) M ( X ) M (Y ), ( ) . . ( X ,Y ) 0 , ( ). , . r( X ,Y ) ( X ,Y ) ( X ) (Y ) M ( XY ) M ( X ) M (Y ) . ( X ) (Y ) (4.2) !), ( 52 . - r( X ,Y ) 1 . 4.2. (X, Y). (X,Y) f ( x, y ) lim x y f1 ( x ) P( x f ( x, y )dy; x) x, y Y x y x 0 0 Y M (Y X X f 2 ( y) y) y . f ( x, y )dx . f ( x, y ) f1 ( x ) f 2 ( y ) . ( y x) f ( x, y ) / f1 ( x ). X y ( y x )dy, (4.3) (X ,Y), f ( x, y ) exp { 1 2 1 r 2 1 2 f ( x, y ) 1 2 (1 r 2 x ( [ ) a1 2 1 ) 5 2r( x a1 1 )( y a2 : a1 , a 2 , , 2 1, ) ( 2, , y a2 2 r. , f ( x, y )dxdy 1; ( x a1 ) 2 1 f 1 ( x) 2 2 1 e ; f 2 ( y) 1 M [( X a1 )(Y 1 2 r 0 f1 ( x ) f 2 ( y ) f ( x, y ) . , : 53 a2 )] ( y a2 ) 2 1 2 r. e 2 2 2 ;, 2 ) ]} . - f ( x, y ) X ; Y– , M (X ) a1 , M (Y ) a2 , (X ) 1, 2; (Y ) r ; ( ). , ( a1 , a2 ), « - » 1 2. : 2 x a1 2r x a1 1 y a2 1 y a2 2 2 const . 2 ( r 0 ), . - 1= 2, . . 4.1 . . 4.1, : 8 f(x, y) 0,15 0,1 0,05 0 a1 6 0 4 - 3; a2 5; 1 1; 2; r 2 0.7. Y - 4 2 . , 6 2 X X Y( !). . 4.1, ) : a1 11 0,04 f(x, y) 0,02 3; a2 2; ( 5 2 Y) Y 2; r X, : 1 -3 3 Y 1 0. - 7 0 5; M (Y X x) a2 r 9 -1 y ( x a1 ); x X ) . 4.1 M (X Y y ) a1 r x y 54 ( y a2 ). X r 1 . - , : y a2 ( x )( x y a1 ) x a1 ( y )( y x a2 ) . 4.3. . , - X . Y. ) , X Y, - XOY. ( ) , nij ). ( X , Y X . - xi , i k 1, k ; Y m nij y j , j 1, m . n i 1j 1 ( ). : m k nxi nij , n yj nij . j 1 x (4.4) i 1 1 k nxi xi , ni 1 1 n y m n yj y j , (4.5) j 1 Y y( xi ) 1 nxi X: m nij y j , i 1, ..., k . (4.6) j 1 ) , (4.1) (4.3). y ( xi ), i 1, ..., k , . rxy xy xy x 55 y , (4.7) x, y– , (4.2). , xy , 1 k m xi y j nij ; ni 1j 1 : 1 k ( xi ni 1 2 x x)2 nxi ; 1 m (yj nj 1 2 y y)2 n yj . (4.2), 1 1. ( . . 0, 1. rxy ), rxy , , - r( X ,Y ) , rxy . X rxy - , - , Y rxy 0, . X Tr rxy . - 0. rxy H 0 : r( X ,Y ) Y, 0; H 1 : r ( X , Y ) 0. - n 2 , 1 rxy2 (4.8) . t 2.cr ( , k ) , . ( Tr - t 2.cr ( , k ) , ); k = n – 2, . 5). ( ( H0 ). ( , (4.7) 2 x , - ). , x - : 1 n 1 n xi ; y yi ; ni 1 ni 1 2 1 n 2 2 xi x ; y ni 1 56 xy 1 n xi yi ; ni 1 1 n 2 yi ni 1 2 y . , X , Y - . 4.2. ( 30 - . 4.1). 4.1 X Y 1 2 3 4 5 6 26 57 36 87 44 35 X Y 77 34 59 25 56 72 7 8 9 10 11 12 19 26 48 33 87 57 X Y 68 67 58 79 16 55 13 14 15 16 17 18 X, 64 98 12 45 5 48 X Y 32 34 78 78 89 35 19 20 21 22 23 24 72 30 18 25 23 78 X Y 42 78 67 57 68 31 25 26 27 28 29 30 25 44 38 52 25 37 68 76 57 38 65 73 0–20, 20–40, 40– . 4.2. Y 60, 60–80, 80–100, . 4.2 X Y 10 30 50 70 90 30 – – 3 9 – 50 – 3 3 2 – 70 – 2 1 – – 90 1 2 – – – nx 4 12 8 3 3 n 30 y ( xi ) 75 65 47.5 36. 67 23.33 – (4.4) (4.6) x ny 10 – – – 3 1 1 7 7 14 1 ny , nx 42.67 , y 54.67 , y ( xi ) . , - , . : xy 1980; : x 43.13; y 57.73; xy 2 x 2 y 519.56; 2120.5; 2 x 57 364.89; rxy , 531.05; 2 y 0.809. 360.93; rxy 0.844. 4.4. , , (X, Y) ( . 4.2, ( ). !) - . , , . : Y ? ( X, 3). – - , , , X ( xi , i X . , 2 y 2 y ( yi y ( xi ) – (3.5) 1 n Y - 1, 2, ..., k ). Y m y )2 n yj (4.9) y ) 2 n xi , (4.10) (yj j 1 1 k ( yi ni 1 ). (3.6). Y, X. Y - (3.10): 2 yx 2 y 2 y . (4.11) ( . yx =0 0 , Y X. rxy yx =1 yx 1( rxy , ). 58 2 yx yx ) 1, - 4.1. 2 y 2 yx 364 .89 0.664 , 2 y (4.10), (4.11) y x 242 .11 , 0 .815 . , , , . 4.5. Y - X 1 , X 2 , ..., X k , A r11 r12 ... r1k r1 y r21 r22 ... r2 k r2 y ... ry1 ... ry 2 ... ... ... ... ryk ryy , (4.12) rij riy A rxi , y . , - rij r ji , riy 1. X 1 , X 2 , ..., X k R D det A , rx i , x j , D yy 1 ryi . , Y D , D yy (4.13) , . 4.3. : X1 X2 X3 Y X1 X2 X3 Y 1 0.3 0.7 0.3 1 0.4 0.5 0.4 0.7 0.5 1 0.2 0.1 59 0.2 0.1 1 . D det A 0.162 , (4.13) R 0.757 . , D yy 0.38 . - , , - 4.6. . ( , , . .), , - . ( . , , . .). : . , . , , « i »). 1, n . xi . , , ( »). (4.7). « y yi . , i. xi i 1, n . . yi xi , - , . (4.8). 4.4. 11 , - : xi 1 2 3 4 5 6 7 8 9 10 11 yi 1 2 3 5 4 9 8 11 6 7 10 . - ? 0.01. 60 rxy : Tr 0.818 , 4.27 , Tr t 2. cr ( 0.01, 9 ) t2. cr , , 3.25 . - . , - , , yi : xi n 6 rxy i 1 1 (4.14) 4.3 6(1 1 9 1 9 9 11 120 1 . n( n 2 1) , rxy yi ) 2 ( xi 9 1) 4.7. 1 240 11 120 9 11 0.818 . . ( ) , ( ) - . . » « - ». X (t ) t , . . , , . . . . , , , - . . , , , . . , . , , , . . , . , , , 61 - , . . , . ( x , . - ), , . « », . « » » « . t , : X (t t) X (t ) . – – . - . x(t ) – xi , i , n 1, n , , , . . ti ( , const . ) . ti 1 . , - – . , , : x1 x2 x … - x 1 x1 … x x2 2 … … (4.7). 2 : – xn , , x xn , - , x1 , - x1 , x2 , ..., xn , , - x xn – ». x 2 ,…, « 1 xn . , , r. (4.7) , 1 r n n xi x 1,n x i 1 i x 1, n (4.15) x 1, n x 62 1, n n 1 xi x n r x i 1,n x 1,n i 1 . x 1,n (4.16) 1,n x , x x - . 1, n . , 2 x , 2 x2 x , (4.16) n 1 xi x n r i x 2 i 1 x 2 x ( . 2 (4.17) , - ) 0. n/3 n/4. (4.15)–(4.17) , r0 . . . , , 1, - – , , , . : 1. 2. « » . r , , ( 1) r0 r1 . - r. , 3. ( , , , r , ) . 4. , , 4.5. . , , F10 .7 , . 11- . 4.2 ( F10 .7 , 10) ). 63 53 ( 1948 2000 . 2500 F10.7 10 2000 1500 1000 500 2000 1996 1992 1988 1984 1980 1976 1972 1968 1964 1960 1956 1952 1948 0 . 4.2. 1948–2000 . . 4.3 : (4.17) – (4.16) – (2). , . 4.2) (1), ( . , , . , r 1 0,8 2 0,6 1 0,4 0,2 0 -0,2 1 2 3 4 5 6 7 8 9 -0,4 -0,6 -0,8 -1 . 4.3. 64 10 11 12 13 14 15 16 : 1. . 4.4. Excel 4.2, - 1 . nx , n y . , (4.5) . (4.6). Y, , (4.9) (4.10) (4.11) - ( - . xy rxy ). , ? ( - ). 3. » , 10 70 70 70 90 30 50 50 50 70 70 70 70 70 70 70 70 70 50 30 30 30 50 50 50 70 70 : : 70 30 30 50 90 10 30 30 X, ; Y , , nij . , , , 3. . , . 65 - 2. 30 X Y 1 2 3 4 5 6 7 8 9 10 26 57 36 87 44 35 19 26 48 33 77 34 59 25 56 72 68 67 58 79 : X Y 11 12 13 14 15 16 17 18 19 20 87 57 64 98 12 45 5 48 72 30 16 55 32 34 78 78 89 35 42 78 X Y 21 22 23 24 25 26 27 28 29 30 18 25 23 78 25 44 38 52 25 37 67 57 68 31 68 76 57 38 65 73 X: 0–2, 2–4, 4–6, 6–8, 8–10; 22, 22–28, 28–34, 34–40; . , 1. ? Y: 10–16, 16– - ? 3. - Y X1 , X 2 , X 3 : X1 X2 X3 Y 4.9 5.3 4.6 5.3 4.8 5.3 4.7 5.3 5.3 5.3 5.3 5.3 4.9 5.5 4.9 5.3 5.4 5.3 5.0 5.0 3.0 3.1 3.4 2.9 3.0 3.1 3.5 2.9 2.8 3.2 3.4 3.3 3.5 2.8 2.9 2.7 2.8 3.0 3.2 2.6 1.5 2.5 0.9 2.2 1.5 2.7 1.1 2.8 2.9 2.0 2.2 1.6 1.4 2.4 2.2 2.5 2.9 2.3 2.1 2.5 9.2 8.6 10.4 8.0 8.5 8.6 10.2 7.9 7.2 9.6 9.7 9.8 10.5 8.2 7.9 7.7 7.8 8.1 8.9 6.8 66 , - . 3 ( ). - (4.13). . ( ). 4. Excel 4.3 : (4.14). 5. 1948–2000 . ( F10.7 1744 1778 1282 1189 854 739 702 950 1846 2327 2317 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1 4.4): 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 F10.7 2097 1621 1054 904 812 726 764 1021 1432 1493 1512 F10.7 1562 1185 1208 934 865 761 734 869 1436 1920 1985 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 F10.7 2026 1753 1196 1011 747 741 852 1409 2137 1896 2082 1992 1993 1994 1995 1996 1997 1998 1999 2000 F10.7 1507 1099 852 772 720 810 1179 1537 1798 16. 5 . , 1778 B2 C3 . . 1744, = B2 , B3 – B. C 1 537. . , B2 . 1- (0.7785) – . 67 - . . 4.2 . 4.3, 1). 1. ? - 2. ? 3. ? 4. ? 5. 6. « ». - ? 5. . . . - . 5.1. , , ( ). - , : x y f (x) x0 y0 x1 y1 … … x2 y2 x xn yn . - . « » , , . . , - , - . f (x) , F (x) , f (x) . , 68 - f (x) F (x) - . F (x) - , . F (x) f (x) . . , F (x) , f (x) . - F (x) , f (x) - F (x) , , , , . , x, , ( x 0 , xn ), . . . , . . ( x 0 , xn ), . y , - , . , y , , . , , x y . - , , . , , , . - , . . . ( ) . - . , , . . - . , , . ( ). . 69 5.2. , y y f (x) f ( x0 ) n 1 : y0 , f ( x1 ) y1 , ..., f ( xn ) yn . , x ), F (x) , ( y, , - . x0 , x1 , ..., xn y f (x) . , y f (x) , . , . , - , F (x) . F (x) , « ( f (x) »). n 1 , n. , Ln ( x ) am (m a0 a2 x 2 a1 x ... a n x n , 0, 1, ..., n ) – . , a0 a0 xi y i (i a1 x0 a2 x0 ... an x0 y 0 , a1 x1 a2 x1 ... an x1 y1 , a0 a1 xn 0, 1, ..., n ) – a 2 xn ... an xn yn , . 1 x0 x02 ... x0n 1 x1 x12 ... x1n .................... 1 xn xn2 70 ... xnn 0, - . , , 0 a0 , : 1 a1 , ..., an n , (m 0, 1, ..., n) 0, 1, ..., n) . m yj (j , n - . : n Ln ( x) y jQ j ( x) . j 0 , Ln ( xi ) Q j (x ) yi . : Q j ( xi ) 0, 1, i i j, j. n (x Q j ( x) xk ) k 0 k j n , (x j xk ) k 0 k j x xj , ) - ( . , : n ( x xk ) n Ln ( x ) j 0 k 0 k j yj n . (x j xk ) k 0 k j 5.1. . 5.1) , : 71 (5.1) x 0 1 2 5 y 0 3 0 15 F ( x) x( x 2)( x 5) x( x 1)( x 2) 0 15 5(5 1)(5 2) 1(1 2)(1 5) 3 x ( x 2)( x 5) x ( x 1)( x 2) 4 4 L3 ( x ) 0 3 1 x ( x 2)(3 x 15 x 1) x( x 2)( x 4) 8 x 6 x 2 4 F ( x) 8 x 6 x 2 x 3. x3 . y 16 14 12 10 8 F(x) = x3 – 6x2 + 8x 6 4 2 x 0 -2 0 1 2 3 4 5 -4 . 5.1.. 5.1. ( 5.1) 5.3. , xi x0 ih, h const (i 0, 1, ..., n) . 1(i yi yi 1 yi . 1- 2(i 0, 1, ..., n 2) : 72 2 yi yi 1 yi . 0, 1, ..., n 1) : n 2- n : y n 1 ( y) . 3- : 2 yi yi 3 yi 1 2 yi yi yi 2 1 yi 2 yi yi yi y ( xi yi 1 2 yi 3 3yi 3 yi 1 yi 2 3yi 2 yi ( yi 1 2 yi 2 2 yi ) y ( xi ) . h yi 1 xi x xi 1 xi xi 1 yi xi , t 1 1 x x xi xi 1 xi F ( x) yi xi , q h t ( yi h yi - ( xi , xi 1 ) , t/h: t yi ) , . . h (5.3) yi q. y ( xi ) , ( xi , xi 2 ) y (x) y ( xi 1 ) h yi 2 2 , y ( xi 2 ) yi 2 yi xi 1 1 2xi 2 xi yi 2 yi . xi 1 , t x (t h )(t 2 h ) 2h 2 ( yi yi ) F ( x) (5.3) xi , q t / h , yi , 2 yi F ( x) yi (5.2) yi . 1 , F ( x ) L1 ( x) 2 yi xi 1 ) x y ( xi 1 ) yi ; 1 L2 ( x ) t ( t 2h ) ( h2 ) yi ( yi yi q 2 yi 2 yi 2 yi ) t (t 2 h ) , 2h 2 q( q 1) . 2 (5.4) (5.4), , - . 5.2. x , , : x 0 2 4 6 y 1 4 2 8 73 2, h x1 F (3) ; 2, t y1 3 2 1, q 1 / 2, y1 y x 3 . 4, y 2 2, y3 8. y1 y2 y1 2 (5.2) 2 y1 y3 2 y2 y1 8 . y1q 4 2 0.5 3. (5.3) F ( 3) y1 (5.4) q( q 1) y1q 2 y1 4 2 0.5 8 0.5 0.5 / 2 2 . 2 . x0 0, y0 1 . 5.4. , , . - , , , . , , yx yx , - f ( x, ) , Y, X = x, – . , , X - Y ( ), yx ax b . , yx . n S( ) ( yi f ( xi , )) 2 . i 1 , , n S ( a , b) ( yi i 1 74 axi b) 2 f ( x, ) a * b* a * x i b* - . y x ( xi ) yi . a* xy x 2 xy ( x) 2 y x2 , b* x 2 x xy ( x) 2 y a* x . (5.5) (4.7) (5.5), , a* xy xy y 2 x rxy , x a * x b* yx yx y y rxy ( x x) . (5.6) x 5.5. X - Y? , . , ( . , . 3.3, , - ). , . 4.4, , . , . ( ). . . Y y 75 - 1 n D 1 n 1 n y x )2 ( yi 1 n y)2 ( yi ( yi yx 2 n y)2 (yx y )2 yx ( yi y x )( y x y ). , . , , - Y , – Y, X - ). n, ( SST SST ( yi y)2 – SSE (yx y)2 – SSR ( yi y x )2 – ). c (3.9)) SSR SSE , (5.7) ; , ( R2 , 0 ; SSE SST . R2 (5.8) R2 , 1. 1 , . : R 2 y )2 y)2 (yx ( yi ( a* ) 2 ( xi x )2 ( yi y ) 2 ( a* ) 2 2 x 2 y rxy2 , . . ( . 4.4, ). 5.6. , , Y . X 1 , X 2 , …, X k , , 76 , Y ( ). a1* x1 y x1 , x 2 ,..., x k , a 2* x2 ... a k* x k b* . (5.9) , . ( » , Excel). R 2 ( y x1 , x 2 ,..., x k SSE SST y )2 y)2 ( yi . (5.10) , , - . : F R2 R( . . 4.5). : F SSE SSR : m n m 1 R 2 ( n m 1) , (1 R 2 )m m (5.11) x ). ) F Fcr ( , m, n m 1) ( . 7). Fcr , F . 5.3. 20 SSE 48.4 , , ), ). SSR 5.6 R2 , R, 0.05 . R 2 SSE / SST SSE /(SSE SSR) 0.896; R F 46.1, Fcr ( 0.05, 3, 16) 3.24. F Fcr , . . 77 R2 0.947 ; - . Y, , . , - , 5.7. , , , - , Y a1 X 1 a2 X 2 b. ak X k ... ; Y , X 1 , X 2 , ..., X k . a1 , a 2 , ..., ak , b , - . , - . , Y a1e X1 a2 X 2 a3 X 32 b. , . U1 e X1 , U2 X 2, U3 X 33 , , Y : a3U 3 b . a1U1 a 2U 2 , . . Y bX 1a1 X 2a 2 e a 3 X 3 , . . , . , ? .( , ). ln Y - ln b a1 ln X 1 a2 ln X 2 78 a3 X 3 . , - Z lnY , c ln b , U ln X 1 , V X3, W :Z a1U a2V a3W c. ln X 2 , , - , * * * * ec x1a1 x2a2 ea3 x3 . y x1 , x2 , x3 , . . , - , . : 1. 4.1 4 : F ( x) L3 ( x ) 8 x 6 x 2 3 x . . 1 1. ( 4.1 - ). ) . ( ) - . , 3, (R^2). , , 4 , , ) ( , R2 1. ( ). Y V 2. a1 X a2 X 2 (5.8) a3 X 3 b X 3. Y, U X2 , - X, U – V. , ( x ji , j 1, k , i 1, n ); A ; B ). yi , i 1, n ; 79 – – Y; , , , (1) ; B– , (1) ; A– (0) - , (0). k 1, 5 k – . . - : a k* sa k a *k 1 sa k 1 … R2 s n 1 k SSR F SSE – a 2* sa 2 a1* sa 1 b* sb – – – – – – – – – – – – Y, s . , , - , , . 5 ). , ,4 4 (5 . . , <CTRL> + <SHIFT> + <ENTER>. , 1( – , , OK. F2, , ). - . 2. 3 ( , - . . 2 . : 80 4), –0,89144 0,19837 0,957746 120,8868 20,8712 2,832728 0,30429 0,239896 16 0,920804 1,005391 0,356699 –3,25573 1,672071 , y x1 , x2 , x3 1,005 x1 2,833 x2 R2 0.891x3 3,256 0.958 , R, 3( F- 4). - . 3. , 20 . 5.1). 1. , Y X1 X2 Y a1* x1 y x1 , x 2 - a1 X 1 a 2* x 2 a2 X 2 b, b* . 2. , bX 1a1 e a 2 X 2 , Y : ln Y ln b a1 ln X 1 Z a2 X 2 ; ln Y , c ln b , U Z a1U ln X 1 , V a 2V 5.1 Y X1 X2 0.51 0.02 1.95 0.65 1.21 2.06 1.67 0.02 0.24 2.82 1.14 0.77 0.02 2.33 1.61 0.02 0.32 1.27 0.28 0.74 0.60 0.15 0.76 0.47 0.80 0.78 0.75 0.11 0.48 1.00 0.67 0.56 0.15 0.99 0.80 0.11 0.36 0.95 0.66 0.92 0.28 0.94 0.12 0.29 0.69 0.22 0.26 0.20 0.88 0.44 0.56 0.29 0.99 0.45 0.19 0.75 0.01 0.93 0.97 0.71 X2; c. a1* , a2* , * 81 * * * ec x1a1 ea2 x2 . y x1 , x2 3. . 4. 20 Y y x1 ,x 2 , - : 1 n 1 n yi y x1 , x 2 ; yi y x1 , x 2 i yi i . 3 . Z, U, V . .2 1. ( ) 2. . 1 , 2, 4. ? 3. 4. ? ? 5. . 6. . 82 6. . . - . . . , , . 6.1. , , - . , . , - , . , . , . . , - . . . 30- . XX . - . . 50- . . - . , , . - . . , , , , , . 83 – , . : , – ; , – - ; - – . , , . , , , ( , . . - ), , « x1 , x2 , ..., xn , » ( . 6.1), ( ) . . 6.1. « » . , xi ( i 1, ... , n ) - . xi . - , . , – - . . , , . - : . , . 6.1 ( y, . 84 ) - . . 6.2. , . . . , , . . , , . . . , , , . - . . , , . . - - . . , : 1. . . , , . , , . 2. - . 3. , - , . , , , , , . 4. . 5. , , . ( - ). 6. , , . , 85 - , . ( ), . – ( ) . - . ( , ), ), ( , , . . , . . - . ( , , ; , . .). , . – , . ( , - , , . 2.3). , , - . , . . . - . ( ), , , . , . . , - . 6.3. , , : – , , ; – , . . ; 86 - – , ); ( – – – - , ; . . , , . , - , n y xi , x j – 0 n i 1 i xi n i, j 1 ij xi x j i 1 ( i 1, ..., n ; j 1, ... , n ; i 2 f ; xi i ij ii f 2 xi n n bi xi i 1 n i, j 1 (6.1) bij xi x j . 5) . (6.1) , y b0 ... , – 2 ( b 2 j ); 2 f ; xi x j ii xi bii xi 2 ... , (6.2) i 1 ( - y ). , . . , . - . . – , , . ( ) - . , . . , - , , 87 ( ) pk . k. , 46 4096 , ! , , - . , ? . , - , : n y b0 n bi xi i 1 n i, j 1 i j bij xi x j i , j ,u 1 i j u (6.3) biju xi x j xu ... , : n y , b0 n bi xi i 1 bii xi 2 i 1 n i, j 1 i j (6.4) bij xi x j ... , - . . 6.4. . , zi (i = 1,…, k) y. - , . . x10, x20, ..., xi0, ..., xk0, ) ), – , . . - ( ), ( 88 ). , +1, –1, - – 0. , - . , - ( ). k n = 2, 2k ( ) k – . 2k). , – , . , , – – , . - . , , - . . zi , zi zi . . zi – zi , zi , i=1,…, k). xi - zi zi0 zi xi , (6.5) i zi0 , zi zi0 –1 zi 2 +1, . . xi – i . zi , i zi 2 - . 1, 1 , i=1,…, k. –1, – +1. : y b0 b1 x1 b2 x2 89 bk xk . (6.6) , y b0 b1x1 b2 x2 bk xk b12 x1x2 b13x1x3 bk 1,k xk 1xk (6.7) , - . 6.5. ( ) - , , – . ) 22 . 23 . 6.1, . 6.2. 6.1 1 2 3 4 x1 –1 +1 –1 +1 x2 –1 –1 +1 +1 ( . 6.2, ), 6.2 1 2 3 4 x1 –1 +1 –1 +1 x2 –1 –1 +1 +1 x3 –1 –1 –1 –1 5 6 7 8 x1 –1 +1 –1 +1 x2 –1 –1 +1 +1 22 ( . 6.2, ). 3 2 – x3 +1 +1 +1 +1 - . 6.2. x1 x2 , 1. , . 90 , n > 3, , , . - , – . , , - . . . : x1 a, x2 . . b, ... , , . (1). - 22 ( . 6.1) : (1), a, b, ab , , , ). 2 3 , . 6.2) : (1), a, b, ab, , a , bc, abc. ( ), , . . : . - : , . . . , , – 4, – 8 . . . 23 . 6.2). , . . «+» «1» «–» «–1»). : 1) , . . ( n i 1 xij 91 0; ) 0: 2) ( n): n xij2 n; i 1 3) ( ): n i 1 0. xij xip (6.8) 6.6. , , . , , , , - . . - , , , . . 2 2 2. , 2 2. +1 –1. ( 2 - 2 , ) n = 2 = 4. , , . : y b0 b1 x1 b2 x2 b12 x1 x2 b11 x12 b22 x22 . 22 6. - . 6.3. 6.3 0 1 2 3 3= 4 x4 – x 22 x5 y 0 1 2 1 +1 –1 –1 +1 +1 +1 y1 2 +1 +1 –1 –1 +1 +1 y2 3 +1 –1 +1 –1 +1 +1 4 +1 +1 +1 +1 +1 +1 y3 y4 92 1 2 5 x12 x0, +1. ( . . - ) , , , . , . , , , , 0. 2k . . , , . . , , 2, . . n. , k 2k , , - . , , , . k 2 , n, , , n. 2 3 n = 3. . . n = 23 = 8 , . 2 2 2 . 6.4. 3 , , , . , , , . , . , , , , . 93 , - 6.4 - –1 y1 y11 y2 y12 y3 y13 +1 +1 y21 y12 y23 +1 –1 +1 y31 y32 y33 +1 –1 –1 –1 y41 y42 y43 +1 +1 –1 –1 +1 y51 y52 y53 –1 +1 –1 +1 –1 –1 y61 y62 y63 –1 +1 +1 –1 –1 +1 –1 y71 y72 y73 +1 +1 +1 +1 +1 +1 +1 y81 y82 y83 0 1 2 3 1 2 1 3 2 3 1 +1 –1 –1 –1 +1 +1 +1 2 +1 +1 –1 –1 –1 –1 3 +1 –1 +1 –1 –1 4 +1 +1 +1 –1 5 +1 –1 –1 6 +1 +1 7 +1 8 +1 1 2 3 y y1 y2 y3 y4 y5 y6 y7 y8 j( - ) m . yj1, yj2, …, yjm , 1 m y ji . mi 1 yj (6.9) , , . . – . . , 2 3 - 3 6 : 2 5 1 7 4 8 4 1 8 5 3 2 7 6 , 3, – 6 . . 1 (j = 1,…, n) - 4, – . . j. , j- ( 94 . 6.4). 6.7. m m 2 s yj i 1 y ji yj m 1 2 (6.10) . . 2 s yj max , . . Gp s2 y j n max , (6.11) 2 j 1 s yj , . Gp 1/n , ( n – ). G ( . 8), f1 - f2 ( ): f1 = m – 1; f2 = n. f1 ), ( f2 ) . Gp < G , ( 1– ) . j - . , ( , , - .). , s 2y , . ( . . 95 - s 2y ) - : n s 2y n m 1 n m 1 n– m– yji – yj – y ji j 1i 1 j 1 2 yj s2 y j , n (6.12) ( ); ( ) ; i j ; j , - (6.9). 6.8. , - , . , (6.6) . , : y b0 b1 x1 b2 x2 . n yj 2 yj min . b i j 1 n b1 2 yj b0 b1 1j b2 2j 1j 0; yj b0 b1 1j b2 2j 2j 0; j 1 n b2 2 j 1 n b0 2 yj b0 b1 1j b2 0. 2j j 1 , n n yj j 1 1j b0 n 1j j 1 n 2 1j b1 x j 1 96 b2 1j j 1 2j 0; n n yj n b0 2j j 1 n b1 2j 1j j 1 2j j 1 n yj 0; j 1 n n b0 n b1 j 1 2 2j b2 b2 1j j 1 2j 0. j 1 : –( k ) –( k ) 0; xij j 1 x 2 ij n; j 1 k –( ) x ji x 0. i i 1 n yj 1j b1n 0; yj 2j b2 n 0; j 1 n j 1 n j 1 yj 0. b0n : yj yj 1j j 1 b1 n n n b2 ; n yj 2j j 1 b0 ; n j 1 n . , (6.6) y . , - , (6.7) ( b0 br , p 1 n , ) . .: bi 1 n n j 1 1 n x0 j y j , n j 1 xij y j , i =1, … ,k; (6.13) n x rj x pj y j , r p, r 1,..., k , p 1, ..., k . ., j 1 . b0 , , x0 , +1. 97 - , , ) , (6.13) - . ( ) . - , . , – , - . , . , , - , , . ( ), - . , . , . 6.1. 2 2, . . y1 = 6, y2 = 3, y3 = 4, y4 = 7. 22 : 0 1 2 +1 +1 +1 +1 –1 +1 –1 +1 –1 –1 +1 +1 , 1 2 ~ y y +1 –1 –1 +1 - 6 3 4 7 y 4,5 4,5 5,5 5,5 y y 6 3 4 7 0 0 0 0 , - x0 , . ~ y b0 b1 (6.13) b1 x1 b2 x2 : b0 1 6 3 4 7 5; 4 1 6 3 4 7 0; 4 98 b2 ~ y ~ y 1 4 6 3 4 7 0,5 . 0,5 x2 . 5 0 x1 . ~ y, y , . , y b0 b1 x1 b2 x2 b12 x1 x2 . . (6.13): b12 1 6 3 4 7 1,5. 4 y 5 0 x1 0,5 x2 1,5 x1 x2 . b12 y ). , ) ( , , , . ( , . , , , . 6.9. . , (6.13). - . t : . ( ) - . ( ) - . tk - bk s : tk bk s 99 . t cr , tk , bk . bk , tcr s - bk t cr s . - b = t cr s . tcr . 5). n– ( n(m – 1), , bk . - (n = 2k)…, m – b ), bk . tk t cr ( ; tk . t cr , , - bk , s s 2y (6.12) s 2y s m n . (6.14) , . - . 6.10. : F . <F , – , F s2 , s 2y F s 2y – s2 , – , (6.15) (6.12), - : 100 s m n n r j 1 2 n– m– r– yj– 2 yj yj , (6.16) ; ; ; , j- ; yj– ( j- - (6.9)). F . 7) ( k1= n – r k2 = n(m – 1). k1 (6.15) – s2 , k2 – s 2y . (6.15) – , . , ; - – . , , , . , , . . - , . , . , ). . , « » , , »– zi . xi 101 - (6.5). - , . , , , , . 6.2. 3 2, y . ), 2 ); z2 – , 2 : z1 – ( ); z3 – 2 ( ). . 6.5. 6.5 2 z1 ) 0,04 0,02 0,06 0,02 z2 ) z3 180 120 300 60 2 ) 5 3 8 2 , - , . : (6.5): 1. xi zi0 zi , i zi0 – ( x1 ), z1 0,04 , 0,02 x2 i – z2 180 , 120 . x3 z3 5 . 3 2. . . 6.6. 102 (6.9) - 6.6 y1 y2 y3 y4 y5 –1 9,38 8,78 9,38 8,79 9,41 9,148 –1 –1 11,69 11,83 11,97 11,41 11,81 11,742 –1 +1 –1 10,44 10,19 10,59 10,28 10,34 10,368 +1 +1 +1 –1 13,35 13,02 13,34 13,05 13,11 13,174 5 +1 –1 –1 +1 11,12 11,01 10,89 11,22 11,19 11,086 6 +1 +1 –1 +1 14,57 13,93 14,35 14,50 14,46 14,362 7 +1 –1 +1 +1 12,66 12,66 12,57 12,73 12,50 12,624 8 +1 +1 +1 +1 15,69 15,97 16,09 15,75 15,50 15,800 0 1 2 3 1 +1 –1 –1 2 +1 +1 3 +1 4 3. y (6.10) : s2(y1) = [(9,38 – 9,148)2 + (8,78 – 9,148)2 + (9,38 – 9,148)2 + + (8,79 – 9,148)2 + (9,41 – 9,148)2]/4 = 0,1100; s2(y2) = [(11,69 – 11,742)2 + (11,83 – 11,742)2 + (11,97 – 11,742)2 + + (11,41 – 11,742)2 + (11,81 – 11,742)2]/4 = 0,0443. s2(y3) = 0,0237; s2(y4) = 0,0254; s2(y5) = 0,0185;s2(y6) = 0,0647; s2(y7) = 0,0080; s2(y8) = 0,0544. (6.11), : Gp = 0,11/(0,1100 + 0,0443 + 0,0237 + 0,0254 + 0,0185 + 0,0647 + 0,0080 + 0,0544); Gp = 0,3151. ( . 8) f1 = 5 – 1 = 4; f2 = 8, = 0,05 = 0,3910; G > Gp, . . , . . . (6.12) - s 2y 8. 103 0,043628 . 4. y ( . 6.7). 6.7 y y - 0 1 2 3 1 +1 –1 –1 2 +1 +1 3 +1 4 1 2 3 y y 1 2 1 3 2 3 –1 +1 +1 +1 –1 9,148 9,0545 –1 –1 –1 –1 +1 +1 11,742 11,7545 –1 +1 –1 –1 +1 –1 +1 10,368 10,4615 +1 +1 +1 –1 +1 –1 –1 –1 13,174 13,1615 5 +1 –1 –1 +1 +1 –1 –1 +1 11,086 11,1515 6 +1 +1 –1 +1 –1 +1 –1 –1 14,362 14,3775 7 +1 –1 +1 +1 –1 –1 +1 –1 12,624 12,5585 8 +1 +1 +1 +1 +1 +1 +1 +1 15,800 15,7845 5. (6.13): b0 = (+1·9,148 + 1·11,742 + 1·10,368 + 1·13,174 + 1·11,086 + 1·14,362 + + 1·12,624 + 1·15,8)/8; b0 = 12,288; b1 = (–1·9,148 + 1·11,742 – 1·10,368 + 1·13,174 – 1·11,086 + 1·14,362 – – 1·12,624 + 1·15,8)/8; b1 = 1,4815; b2 = (–1·9,148 – 1·11,742 + 1·10,368 + 1·13,174 –1·11,086 – 1·14,362 + + 1·12,624 + 1·15,8)/8; b2 = 0,7035. b3 = 1,18; b12 = 0,014; b13 = 0,1315; b23 = 0,0405; b123 = –0,039. 6. . S (6.14). S y2 S 0,043628 8 5 m n 0,033026. = 0,05 ( f n m 1 8 4 . 5) 32 104 - tcr 2,04 . , 2,04 0,033026 0,067372. b t cr S - , b23 b123 , 0,067372. , b12, , , . , y 12,288 1,4815 x1 0,7035 x2 1,18 x3 0,1315 x1 x3 . 7. . , S F (6.16). yj, 2 - j, . 6.7. y1 12,288 1,4815 y2 0,7035 12,288 1,4815 1 0,7035 y3 y4 1 12,288 1,4815 13,1615 ; y5 1 1 1 1,18 1,18 1 0,7035 1 1,18 11,1515 ; y6 1 14,3775 ; y7 0,1315 1 9,0545 ; 0,1315 1 11,7545 ; 0,1315 1 10,4615 . 1 12,5585 ; y8 15,7845. : S 2 m n n r j 1 2 yj yj 5 (8 5) [(9,0545 – 9,148)2 + (11,7545 – 11,742)2 + +(10,4615 – 10,368)2 + (13,1615 – 13,174)2 + (11,1515 – 11,086)2 + (14,3775 – –14,362)2 + (12,5585 – 12,624)2 + (15,7845 – 15,8)2] = 0,044763. F F 0,05 f2 = n(m – 1) = 8 4 = 32 : F s2 s 2y 0,044763 1,026 . 0,043628 F ( . 7) f1 = n – r = 8 – 5 = 3 (0,05; 3; 32) = 2, 901. 105 F . = 1,026<F = 2, 901, - 8. : y 12,288 1,4815 x1 0,7035 x2 1,18 x3 0,1315 x1 x3 . , 1 – , . ( ) 2 3 – – ( 1 3 : . - ) – . , , ( ) . 9. , (6.5) - zi : xi y 12,288 1,4815x1 0,7035x2 1,18x3 0,1315x1x3 12,288 1,4815 z1 0,04 z 180 z 5 z 0,04 z3 5 . 0,1315 1 1,18 3 0,7035 2 0,02 3 3 120 0,02 , - : y 6,7414 63,1167 z1 0,0059 z2 0,3057 z3 1,8858 z1 z3 . : 1. 6.2. SExcel. 1 ( Excel A2:F9 . 6.3). ( F2:F9). 106 . 6.3. , , G2:G9. , . . 8 ( G13). G14, , . . 8. G10 Excel . 6.7 ( y ( . 6.4, J17:K24 ). (6.13) A27:H27. - A17:I24), - , - G29 S (6.14), . ( tcr . 5) - 2,04 b tcr S . G31. b , 107 b A34:H34). . 6.4. . - yj. J17:J24 , ( js2 A34:H34) ( A17:H17 A24:H24). (6.16) , - , , , , ( K17:K24). . F J29, F K25. (6.15) .7( - J32). F = 1,026 < F . 2. (U – = 2, 901, - (z1), I – 108 (z2), T – (z3)) . - : U = 30 , I = 18 , T = 220 °C. 2 3 . . 6.8. 6.8 z1 = U, z2 = I, 30 2 32 28 z3 = T, 220 20 240 200 18 1 19 17 5 . 9. , yi – , . : 1. 2. . . 3. 4. 5. . 6. . . . , , . , . 7. 8. . . S Excel 2 1. 1. ? . 2. 3. 4. 5. 6. ? . . ? ? ? 109 - 7. ? 8. ? ? 9. - 10. - ? ? 11. 12. ? , yi , , , ? 13. - ? F ? 14. ? 7. . - . . 7.1. , (n > 6), . 2 - n 7 7 2 = 128 , . . , - ( , 7 N 1 C72 C73 C74 C75 C76 1 128 . b0 b2 x2 ) , y b1 x1 , bk xk , 1+7 = 8 ; 1 + 7 + 21 = 29. , 2 n . , , « » 110 - , - . . , ( - ). , - , . . , . , , - . (6.13) , . 23, 2, 3, 5, 8, . 7.1, : . 7.1 x1 –1 +1 –1 +1 1 2 3 4 x2 –1 –1 +1 +1 x3 –1 –1 –1 –1 x1 –1 +1 –1 +1 5 6 7 8 x2 –1 –1 +1 +1 x3 +1 +1 +1 +1 , x2 x1 , 2 -1 - 2 =2 . x3 x3 x1 x 2 , (7.1) x3 x1 x 2 . (7.2) – . ( 2 2 ( . 7.2), x3 , , ) , x1x2. 111 - 7. 2 x1 1 (2) 2 (3) 3 (5) 4 (8) x2 x3 +1 –1 –1 +1 x1 –1 +1 –1 +1 –1 –1 +1 +1 1 2 3 4 x2 +1 –1 –1 +1 , x3 –1 +1 –1 +1 . 7.1. 22 , x1x2 –1 –1 +1 +1 - x3. . (7.1) (7.2) - , . , , , - . 2k-p, p– . k 2. , p=1 , (1/2- ) 26-1 n = 26-1 = 32, - . p=1 6-2 6-2 2 n = 2 = 16 ). p = 2 (1/47.2. , , . , . , - , , . , . 2k-p : 1. . 2. l=k–p 2l , . . 1, 2, …, l. 3. 2k-p. 112 2l , l+1, l+2, …, k, p = k – l. , , . . : 1. y . . k=3 b0 b1 x1 b2 x2 b3 x3 b12 x1 x2 k=3 b13 x1 x3 8 : b23 x 2 x3 . p = 1, . . 2. 1 22 ( )( b123 x1 x 2 x3 , n = 23-1 = 4 . 2, . . l = k – p = 2. - . 7.3). 7.3 1 2 3 4 3. x3 = x1x2. . . 1 2 –1 +1 –1 +1 –1 –1 +1 +1 x3 = x1x2 +1 –1 –1 +1 , 3 1 2. , . 7.3 ( , - . 7.4). 7.4 1 2 3 4 0 1 2 3 +1 +1 +1 +1 –1 +1 –1 +1 –1 –1 +1 +1 +1 –1 –1 +1 8, , 7, 2– 1 2 1 3 2 3 1 2 3 +1 –1 –1 +1 –1 –1 +1 +1 –1 +1 –1 +1 +1 +1 +1 +1 1 3– 6, 4– 5. , . - , . 113 , b0 , b1 , b2 , b3 , , : y b0 b3 x3 , b1 x1 b2 x2 i, ( ). , (8) (4). - , - . « . » , . : 23, 2– 13, 3– b0 = 0+ 12, 123, 0– b1 = 123 ( 1+ 1 . 23 , . 7.4). b2 = 2+ 13, b3 = 3+ 12, – . , bi 1 n , - n j 1 xij y j . , , . , x4 24-1 , , . . x4 x1 x 2 x3 . , x1 x 2 , x4 , x1 x 3 , x 4 . , ( x2 x 3 , - ). 7.1. ( [5]) - . 25-2 114 . - , . 5: 4 . : z1 – 3 , , 3 ( . 7.1. 3 , 3 ; z3 – ; z4 – ,% , ; z2 – ; z5 – . 7.1). (z1, ..., z5) (y) . 7.5. 7.5 z1, ( ) 3 z2, 5250 4000 6500 1250 3 3 z3, 3900 3100 4700 800 3 z4, 2650 1750 3500 900 z5, % 1100 700 1500 400 74 50 98 24 (6.5): zi0 zi xi , (7.3) i zi0 – x1 z1 5250 , x2 1250 – zi 2650 , x4 900 ( z2 3900 , x3 800 ), : z i 74 . 24 i zi 1100 , x5 400 zi , , i = –1. 115 i = +1, - 25 = 32 , . . 1/4- . 25-2, - 2 x4 = x1x2, x5 = x1x2x3. 3 2 = 8. , » . , ( . . 7.6 ) - 23, – - ( ). 7.6 3 2 1 0 1 2 3 4 5 +1 +1 +1 +1 +1 +1 +1 +1 +1 –1 +1 –1 +1 –1 +1 –1 +1 +1 –1 –1 +1 +1 –1 –1 +1 +1 +1 +1 –1 –1 –1 –1 +1 –1 –1 +1 +1 –1 –1 +1 +1 –1 –1 +1 –1 +1 +1 –1 2 yi y1 y2 yi –0,6 0,1 0,6 –0,1 0,6 –0,2 0,1 0,3 –0,5 0,5 0,4 0,2 0,2 –0,2 0,2 0,3 –0,55 0,30 0,50 0,05 0,40 –0,20 0,15 0,30 –0,3750 0,2625 0,5375 –0,1250 0,2250 –0,1625 0,1125 0,4750 , - 6: 1. : yj y j1 y j 2 ... yjm m m=2– . y1 ( 0,6) ( 0,5) 2 , , 0,55. ( 2. ( m ( y ji 2 s yj y j )2 i 1 m 1 ; 2 s y1 ) . . 7.6). : ( 0,6 ( 0,55))2 ( 0,5 ( 0,55))2 2 1 116 0,005. , s 2 y 2 = 0,08; s 2 y5 = 0,08; s 2 y3 = 0,02; s 2 y6 = 0; s 2 y7 = 0,005; ( 8 s 2 y 4 = 0,045; ) s 2 y8 = 0. : s 2 y j = 0,005 + 0,08 + 0,02 + 0,045 + 0,08 + 0,005 = 0,235. j 1 3. : Gp s 2 yi n max s 2 yi 0,08 0,235 0,34 . i 1 .8 G ( ; f1; f2). G (0,05; 1; 8) = 0,6798. f2 = n = 8 = 0,05, f1 = m –1; Gp < G , - . 4. : n j 1 b0 y j x0 j n 0,55 0,3 0,5 0,05 0,4 0,2 0,15 0,3 8 0,11875; 0,55 0,3 0,5 0,05 0,4 0,2 0,15 0,3 8 0,00625; n j 1 b1 y j x1 j n n b2 j 1 y j x2 j n 0,55 0,3 0,5 0,05 0,4 0,2 0,15 0,3 8 0,13125; 0,55 0,3 0,5 0,05 0,4 0,2 0,15 0,3 8 0,04375; 0,55 0,3 0,5 0,05 0,4 0,2 0,15 0,3 8 0,06875; n b3 j 1 y j x3 j n n b4 j 1 y j x4 j n n b5 j 1 y j x5 j n 5. 0,55 0,3 0,5 0,05 0,4 0,2 0,15 0,3 0,256.25 . 8 . ( ): 117 n 1 n m 1 s 2y n m j 1i 1 y ji yj j 1 2 s2 y j 0,235 0,02938. 8 n : S 2y S 0,02938 2 8 m n 0,04285. tcr n(m –1) = 8·1 = 8 - = 0,05 . tcr (0,05;8)=2,31. - : bj b3 = 0,044 < 0,099 , , 2,31 0,04285 0,099 . t cr S bj b1 = 0,006 < 0,099; , b4 = 0,069 < 0,099. . y = 0,119 – 0,131x2 – 0,256x5. . 7.6. yi 6. . : s m n n r j 1 2 2 ~ yj yj . r – . m = 2; n = 8; r = 3 s2 2 ( 0,55 0,375) 2 8 3 (0,4 0,225)2 (0,3 0,2625) 2 (0,5 0,5375) 2 (0,05 0,125) 2 ( 0,2 0,1625)2 (0,15 0,1125) 2 F 118 (0,3 0,475) 2 ] 0,0512. S2 s 2y F 0,0512 1,7427 . 0,02938 F = 0,05 k2 = n(m – 1) = 8·1 = 8. k1 = m(n – r) = 2(8 – 3) = 10 k1 s 2y . s , k2 – F (0,05; 10; 8) = 3,07. . F < F , (7.3). z2 3900 zi 74 , x5 800 24 y = 0,119 – 0,131x2 – 0,256x5: x2 y 0,119 0,131 z2 3900 800 zi 74 . 24 0,256 ó 1,547 0,00016z2 0,0107z5 . 7.3. , - . ? - . . 7.4. . 1 2. 3 , , . . . . , , . 1 2, 3 3 2 3 3, 12 1 2 1. , - 1 2 3. 1 2. xi 2 , , 1 2 3 119 1. 1, . 1, 2 1 2 3 1 2 3, 1 2, 3 0 . . 2 3. 1 1, 1 3, 2 3, 2, 1 2 3 , 1 2 3. 0 : b0 = 0+ 123, b1 = 1+ 23, b2 = 2+ 13, b3 = , 3+ 12. , . , , 4-1 . 2 4 1 2 3 4 4 1 ), 4. ( , 1, 1 2 4 1 2 1 2 3 - 3. . - . 1 2 3, 4 , , 4 – . 7.7). 1 2 7.7 1 2 3 4 5 6 7 8 0 1 2 3 +1 +1 +1 +1 +1 +1 +1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 –1 +1 +1 –1 –1 +1 +1 –1 –1 –1 –1 +1 +1 +1 +1 4= 2 3 –1 +1 +1 –1 +1 –1 –1 +1 2 4 : 1 2 3 4 b1 1 + 234; 2 1 3 4 b2 2 + 134; 120 1 x5 = 1 +1 –1 –1 +1 +1 –1 –1 +1 1 2 3 4 1. 2 3 1 2 4 b3 3 + 124; 4 1 2 3 b4 4 + 123; 1 2. 4 2 4 , 1. 1 2 4 1 2 4 : 1 2 4 b1 1 + 24; 2 1 4 b2 2 + 14; b3 1 2 3 4 3 b4 1 2 4 3 4 + + 1234; 12. . , . , . , 4= 1 2 3. . 4 - , 1 2 , . , , . . - , . , . 7.4. , , ( - ). , , . , , : 1 1 2 4 5-2 1 4 1 2 2 5 1 2 3 1 2 3 5. ; , , 121 - 2 1= , 1 2 1 2 3 4 5 3 4 5. = : 1 2 4 ,1 1 2 4= 1 1 1 2 3 5 1 2 3 : 5= 3 4 5 . 3 4 5. : x1: 1 2 4 = 2 3 5= 1 3 4 5; x2: 2 1 4 = 1 3 5= x3: 3 1 2 3 4= x4: 4 1 2 x5: 5 1 2 4 5= = 2 3 4 5; 1 2 5= 1 2 3 4 5= 1 2 3= 4 5 ; 3 5; 3 4. b1 1 24 + 235+ 1345; b2 2 14 + 135+ 2345; b3 3 1234 b4 4 12 b5 5 1245 + + + 125+ 45; 12345+ 35; 123+ 34. , . , . . , - 1/4. 7 . 1/427-2, . ( 1/47 , 2, 128 29 32 – 1, –7 – 21). , , . x1, x2, x3, x4, x5 . , , : x6 = x1x2x3x4; 122 - x7 = x1x2x3x4x5. , 1 = x1x2x3x4x5x7 : 1 = x1x2x3x4 x6 . , 1 = x1x2x3x4x5x7 = x1x2x3x4 x6 = x5x6x7. : 1 = x5x6x7, . . : x1: 1= 2 3 4 5 7 x2: 2 1 3 4 5 7= 1 3 x3: 3 1 2 4 5 7= 1 2 4 6 x4: 4 1 2 3 5 7= 1 2 3 6= 4 5 6 7 2 3 4 6= 1 5 6 7; 4 6= 2 5 6 7; = 3 5 6 7; . . b1 1 + 23457 + 2346+ 1567; b2 2 13457 + 1346+ 2567; b3 3 12457 + 1246+ 3567; b4 4 12357 + 1236+ 4567 . . , , , . , , : x1x2 = x3x4x5x7 = x3x4 x6 = x1 x2x5x6x7. 27-2 , , 7 2 . , (6.7). k 2. k=5 1/4- k = 2, 3 4 1/2- k=6 . 2k- 1, - k = 7, k = 8 2k- 2 . 1/4- , , , - , . , , , , 123 - , . : 1. 2 , 6 . : V 5-2 t, 2 . , 6. , z1, z2, z3 - (U – (z1), I – (z2), T – (z4) t – . . 7.8. (z3), V – - (z5)) 5-2 2 . 7.8 z1 = U, z2 = I, 30 2 32 28 . 7.9. z3 = T, 220 20 240 200 18 1 19 17 x4 z4 = V, z5 = t, 10 3 13 7 80 15 95 65 x5 . 7.9 0 1 2 3 1 +1 –1 –1 2 +1 +1 3 +1 y1 y2 y3 4 5 –1 – – 7,87 7,41 8,12 –1 –1 – – 16,23 15,42 15,64 –1 +1 –1 – – 6,55 5,89 6,26 4 +1 +1 +1 –1 – – 8,49 9,21 8,79 5 +1 –1 –1 +1 – – 20,16 19,84 20,59 6 +1 +1 –1 +1 – – 28,34 27,59 28,16 7 +1 –1 +1 +1 – – 8,85 8,20 9,64 8 +1 +1 +1 +1 – – 24,19 23,56 23,04 : y – . , 124 - : 1. . x4 x5 . 2. . 3. 4. 5. 6. . 7. . . . . S Excel . S Excel 1. 1 6. , ? 2. ? 3. , ? 4. ? 5. ? 6. 7. 8. 9. ? . 2 -l? . 125 1. . . / . . , 2014. – 344 . . . .– : . . , : 2. . . 3. : : 4. . . / . . , . . , . . .– : . / , 2011. – 72 . , . , 2013. – 132 . / . . , . .– . : .– : . 6. 7. [ , . . . : . .– . . . ] / . . , 2014. – 77 . .– . .: / . , . . . : , . – 2- ., , 2014. – 495 . : » / 2010. – 90 . 8. . . : 2012. – 185 . - , 2003. – 208 . 5. . . . - / . . , 126 . . . . « . – . – : , : , 1 (z) z 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 (z ) 0.39894 0.39886 0.39862 0.39822 0.39767 0.39695 0.39608 0.39505 0.39387 0.39253 0.39104 0.38940 0.38762 0.38568 0.38361 0.38139 0.37903 0.37654 0.37391 0.37115 0.36827 0.36526 0.36213 0.35889 0.35553 0.35207 0.34849 0.34482 0.34105 0.33718 0.33322 0.32918 0.32506 0.32086 0.31659 0.31225 0.30785 0.30339 0.29887 0.29431 0.28969 0.28504 0.28034 0.27562 0.27086 0.26609 0.26129 0.25647 0.25164 0.24681 1 e 2 z 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84 1.86 1.88 1.90 1.92 1.94 1.96 1.98 z2 2 (z ) 0.24197 0.23713 0.23230 0.22747 0.22265 0.21785 0.21307 0.20831 0.20357 0.19886 0.19419 0.18954 0.18494 0.18037 0.17585 0.17137 0.16694 0.16256 0.15822 0.15395 0.14973 0.14556 0.14146 0.13742 0.13344 0.12952 0.12566 0.12188 0.11816 0.11450 0.11092 0.10741 0.10396 0.10059 0.09728 0.09405 0.09089 0.08780 0.08478 0.08183 0.07895 0.07614 0.07341 0.07074 0.06814 0.06562 0.06316 0.06077 0.05844 0.05618 127 z 2.00 2.02 2.04 2.06 2.08 2.10 2.12 2.14 2.16 2.18 2.20 2.22 2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46 2.48 2.53 2.58 2.63 2.68 2.73 2.78 2.83 2.88 2.93 2.98 3.03 3.08 3.13 3.18 3.23 3.28 3.38 3.48 3.58 3.68 3.78 3.88 3.98 4.20 5.00 (z ) 0.05399 0.05186 0.04980 0.04780 0.04586 0.04398 0.04217 0.04041 0.03871 0.03706 0.03547 0.03394 0.03246 0.03103 0.02965 0.02833 0.02705 0.02582 0.02463 0.02349 0.02239 0.02134 0.02033 0.01936 0.01842 0.01625 0.01431 0.01256 0.01100 0.00961 0.00837 0.00727 0.00631 0.00545 0.00470 0.00405 0.00348 0.00298 0.00254 0.00216 0.00184 0.00132 0.00094 0.00066 0.00046 0.00031 0.00021 0.00014 0.00006 0.00000 2 1 2 ( x) x 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 (x ) 0.00798 0.01595 0.02392 0.03188 0.03983 0.04776 0.05567 0.06356 0.07142 0.07926 0.08706 0.09483 0.10257 0.11026 0.11791 0.12552 0.13307 0.14058 0.14803 0.15542 0.16276 0.17003 0.17724 0.18439 0.19146 0.19847 0.20540 0.21226 0.21904 0.22575 0.23237 0.23891 0.24537 0.25175 0.25804 0.26424 0.27035 0.27637 x 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 (x ) 0.28230 0.28814 0.29389 0.29955 0.30511 0.31057 0.31594 0.32121 0.32639 0.33147 0.33646 0.34134 0.34614 0.35083 0.35543 0.35993 0.36433 0.36864 0.37286 0.37698 0.38100 0.38493 0.38877 0.39251 0.39617 0.39973 0.40320 0.40658 0.40988 0.41308 0.41621 0.41924 0.42220 0.42507 0.42785 0.43056 0.43319 0.43574 128 x e z2 2 dz 0 x 1.54 1.56 1.58 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84 1.86 1.88 1.90 1.92 1.94 1.96 1.98 2.00 2.02 2.04 2.06 2.08 2.10 2.12 2.14 2.16 2.18 2.20 2.22 2.24 2.26 2.28 (x ) 0.43822 0.44062 0.44295 0.44520 0.44738 0.44950 0.45154 0.45352 0.45543 0.45728 0.45907 0.46080 0.46246 0.46407 0.46562 0.46712 0.46856 0.46995 0.47128 0.47257 0.47381 0.47500 0.47615 0.47725 0.47831 0.47932 0.48030 0.48124 0.48214 0.48300 0.48382 0.48461 0.48537 0.48610 0.48679 0.48745 0.48809 0.48870 x 2.30 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46 2.48 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.20 4.40 4.70 5.00 (x ) 0.48928 0.48983 0.49036 0.49086 0.49134 0.49180 0.49224 0.49266 0.49305 0.49343 0.49379 0.49461 0.49534 0.49598 0.49653 0.49702 0.49744 0.49781 0.49813 0.49841 0.49865 0.49886 0.49903 0.49918 0.49931 0.49942 0.49952 0.49966 0.49977 0.49984 0.49989 0.49993 0.49995 0.49997 0.49999 0.49999 0.49999 0.50000 3 t ( , n) n 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 35 40 45 50 60 70 80 90 100 120 0,95 0,99 0,999 2.78 2.57 2.45 2.37 2.31 2.26 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.11 2.10 2.09 2.06 2.05 2.03 2.02 2.02 2.01 2.00 2.00 1.99 1.99 1.98 1.98 1.96 4.60 4.03 3.71 3.50 3.36 3.25 3.17 3.11 3.06 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.80 2.76 2.72 2.71 2.69 2.68 2.66 2.65 2.64 2.63 2.63 2.62 2.58 8.61 6.86 5.96 5.41 5.04 4.78 4.59 4.44 4.32 4.22 4.14 4.07 4.02 3.97 3.92 3.88 3.75 3.66 3.60 3.56 3.53 3.50 3.46 3.44 3.42 3.40 3.39 3.37 3.29 129 4 2 s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.05 0.01 0.001 3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 18.31 19.68 21.03 22.36 23.68 25.00 26.30 27.59 28.87 30.14 31.41 32.67 33.92 35.17 36.42 37.65 38.89 40.11 41.34 42.56 43.77 6.63 9.21 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 24.73 26.22 27.69 29.14 30.58 32.00 33.41 34.81 36.19 37.57 38.93 40.29 41.64 42.98 44.31 45.64 46.96 48.28 49.59 50.89 10.83 13.82 16.27 18.47 20.52 22.46 24.32 26.13 27.88 29.59 31.26 32.91 34.53 36.12 37.70 39.25 40.79 42.31 43.82 45.32 46.80 48.27 49.73 51.18 52.62 54.05 55.48 56.89 58.30 59.70 130 5 ( k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 60 120 ) 0.10 6.31 2.92 2.35 2.13 2.01 1.94 1.89 1.86 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.75 1.74 1.73 1.73 1.73 1.71 1.70 1.68 1.67 1.66 1.64 0.05 12.70 4.30 3.18 2.78 2.57 2.45 2.36 2.31 2.26 2.23 2.20 2.18 2.16 2.14 2.13 2.12 2.11 2.10 2.09 2.09 2.06 2.04 2.02 2.00 1.98 1.96 0.02 31.82 6.97 4.54 3.75 3.37 3.14 3.00 2.90 2.82 2.76 2.72 2.68 2.65 2.62 2.60 2.58 2.57 2.55 2.54 2.53 2.49 2.46 2.42 2.39 2.36 2.33 0.01 63.70 9.92 5.84 4.60 4.03 3.71 3.50 3.36 3.25 3.17 3.11 3.05 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.85 2.79 2.75 2.70 2.66 2.62 2.58 0.002 318.30 22.33 10.22 7.17 5.89 5.21 4.79 4.50 4.30 4.14 4.03 3.93 3.85 3.79 3.73 3.69 3.65 3.61 3.58 3.55 3.45 3.39 3.31 3.23 3.17 3.09 0.001 631.00 31.60 12.90 8.61 6.86 5.96 5.40 5.04 4.78 4.59 4.44 4.32 4.22 4.17 4.07 4.01 3.96 3.92 3.88 3.85 3.72 3.65 3.55 3.46 3.37 3.29 0.05 0.025 0.01 0.005 0.001 0.0005 ( 131 ) 6 W 132 7 Fcr ( , k x , k y ) = 0.01 ky\kx 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 7 8 9 4052 98.49 34.12 21.20 16.26 13.74 12.25 11.26 10.56 10.04 9.86 9.33 9.07 8.86 8.68 8.53 8.40 4999 99.01 30.81 18.00 13.27 10.92 9.55 8.65 8.02 7.56 7.20 6.92 6.70 6.51 6.36 6.23 6.11 5403 90.17 29.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5625 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.20 5.03 4.89 4.77 4.67 5764 99.33 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.69 4.56 4.44 4.34 5889 99.30 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 5928 99.34 27.67 14.98 10.45 8.26 7.00 6.19 5.62 5.21 4.88 4.65 4.44 4.28 4.14 4.03 3.93 5981 99.36 27.49 14.80 10.27 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 6022 99.36 27.34 14.66 10.15 7.98 6.71 5.91 5.35 4.95 4.63 4.39 4.19 4.03 3.89 3.78 3.68 10 11 6056 6082 99.40 99.41 27.23 27.13 14.54 14.45 10.05 9.96 7.87 7.79 6.62 6.54 5.82 5.74 5.26 5.18 4.85 4.78 4.54 4.46 4.30 4.22 4.10 4.02 3.94 3.86 3.80 3.73 3.69 3.61 3.59 3.52 = 0.05 ky \ kx 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 161 200 216 225 230 234 18.51 19.00 19.16 19.25 19.30 19.33 10.13 9.55 9.28 9.12 9.01 8.94 7.71 6.94 6.59 6.39 6.26 6.16 6.61 5.79 5.41 5.19 5.05 4.95 5.99 5.14 4.76 4.53 4.39 4.28 5.59 4.74 4.35 4.12 3.97 3.87 5.32 4.46 4.07 3.84 3.69 3.58 5.12 4.26 3.86 3.63 3.48 3.37 4.96 4.10 3.71 3.48 3.33 3.22 4.84 3.98 3.59 3.36 3.20 3.09 4.75 3.88 3.49 3.26 3.11 3.00 4.67 3.80 3.41 3.18 3.02 2.92 4.60 3.74 3.34 3.11 2.96 2.85 4.54 3.68 3.29 3.06 2.90 2.79 4.49 3.63 3.24 3.01 2.85 2.74 4.45 3.59 3.20 2.96 2.81 2.70 133 7 8 237 239 19.36 19.37 8.88 8.84 6.09 6.04 4.88 4.82 4.21 4.15 3.79 3.73 3.50 3.44 3.29 3.23 3.14 3.07 3.01 2.95 2.92 2.85 2.84 2.77 2.77 2.70 2.70 2.64 2.66 2.59 2.62 2.55 9 241 19.38 8.81 6.00 4.78 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.72 2.65 2.59 2.54 2.50 10 11 242 243 19.39 19.40 8.78 8.76 5.96 5.93 4.74 4.70 4.06 4.03 3.63 3.60 3.34 3.31 3.13 3.10 2.97 2.94 2.86 2.82 2.76 2.72 2.67 2.63 2.60 2.56 2.55 2.51 2.49 2.45 2.45 2.41 8 G n = 0,05 f = m –1 1 2 3 4 5 6 2 0,9985 0,9750 0,9392 0,9057 0,8772 0,8534 3 0,9669 0,8709 0,7977 0,7457 0,7071 0,6771 4 0,9065 0,7679 0,6841 0,6287 0,5859 0,5598 5 0,8412 0,6338 0,5991 0,5441 0,5065 0,4783 6 0,7608 0,6161 0,5321 0,4803 0,4447 0,4184 7 0,7271 0,5612 0,4800 0,4307 0,3974 0,3726 8 0,6798 0,5157 0,4377 0,3910 0,3595 0,3362 9 0,6385 0,4770 0,4027 0,3584 0,3286 0,3067 10 0,6020 0,4450 0,3733 0,3311 0,3029 0,2823 12 0,5410 0,3924 0,3264 0,2880 0,2624 0,2439 16 0,4709 0,3346 0,2758 0,2419 0,2195 0,2034 20 0,3894 0,2705 0,2205 0,1921 0,1935 0,1602 134 9 7 1 3, 31 5, 52 3, 50 7, 99 4, 97 9, 23 6, 03 14, 80 2 3,51 3,05 6,47 6,14 4,61 4,93 9,15 9,66 5,28 5,47 13,41 13,81 8,64 8,71 25,75 25,63 3 2,24 2,50 3,82 3,94 2,05 2,52 4,07 4,42 3,07 3,89 5,81 5,48 3,48 3,01 6,83 6,20 4 2,88 2,97 4,91 4,65 3,01 3,31 5,09 5,53 3,73 3,30 6,18 6,52 4,63 4,96 9,38 9,53 5 3,72 3,28 5,93 5,59 3,32 3,55 7,94 7,26 4,40 4,04 9, 63 9,17 6,36 6,96 14,76 14,68 3, 04 5, 93 3, 27 7, 41 4, 84 9, 35 6, 71 14, 72 3, 35 5, 77 3, 36 7, 11 4, 88 9, 66 6, 43 14, 95 3, 39 3,21 5, 09 5, 51 3, 98 3,89 7, 38 7,07 4, 30 4, 29 9, 34 9, 17 6, 39 6, 37 14,68 14,67 6 4, 92 4, 85 8, 85 8, 90 5, 81 5, 86 13, 49 13,32 7, 89 7, 68 20, 52 20,71 11, 82 11,69 66,71 66,13 7 4,07 4,84 8,87 8,96 5,32 5,73 13, 29 13,04 7,79 7,15 20,55 20,78 11,26 11,38 66,99 66,05 8 3,83 3,05 6,55 6, 64 4,75 4,90 9,04 9,30 5,55 5,89 13,40 13,11 8,38 8,31 25,86 25,44 9 3,54 3,32 5,47 5,70 3,26 3,95 7,17 7,21 4,01 4,82 9,50 9,19 6,90 6,83 14,77 14,70 10 2,49 2,37 4,18 4,64 3,36 3,20 5,45 5,45 3,25 3,12 6,21 6,14 4,51 4,89 9,35 9,93 3,53 3,92 3,27 6,35 6,62 6,81 4,16 4,92 4,01 9,34 9,52 9,28 5,42 5,05 5,86 13,51 13,89 13,63 8,38 8,65 8,66 25,11 25,78 25,34 2,39 3,68 2,55 4,76 3,96 5,23 3,14 6,32 2,10 3,74 2,74 4,17 3,19 5,92 3,51 6,58 2,57 3,72 2,73 4,55 3,37 5,73 3,19 6,69 2,42 4,52 3,02 5,48 3,80 6,60 5,01 9,02 2,37 4,29 3,99 5,11 3,89 6,09 4,52 9,46 2,39 4,38 3,48 5,45 3,52 6,43 5,07 9,77 3,80 3,49 3,69 5,63 5,20 5,68 3,93 3,15 3,39 7,49 7,02 7,58 4,68 4,98 4,24 9,60 9,33 9,91 6,69 6,05 6,57 14,25 14,22 14,41 135 4, 33 8,44 5, 47 13, 57 7, 39 20,27 11, 93 66,62 4,24 8, 21 5,90 13, 42 7, 19 20, 58 11, 49 66,85 4, 77 8,39 5,69 13, 56 7, 42 20,31 11,54 66,20 4,28 8,30 5,86 13,28 7,45 20,04 11,71 66,88 4,16 8,89 5,43 13,40 7,68 20,79 11,34 66,55 4,66 8,04 5,62 13,12 7,38 20,61 11,73 66,62 3,23 3,23 3,87 6,23 6,59 6,11 4,76 4,98 4,44 9,24 9,57 9,30 5,27 5,81 5,63 13,45 13,61 13,36 8,03 8,19 8,10 25,78 25,62 25,56 3,024 3,46 3,19 5,78 5,90 5,77 3,37 3,31 3,15 7,01 7,30 7,91 4,90 4,18 4,19 9,15 9, 62 9,36 6,84 6,78 6,78 14,18 14,90 14,93 2,63 4,55 3, 02 5,49 3,90 6,41 4,55 9,05 2,64 4,26 3,12 5,72 3,82 6,04 4,41 9,11 2,69 4,51 3,07 5,55 3,81 6,22 4,81 9,26 16 8,46 12,52 10,05 15,82 11,51 17,31 14,06 22,74 11 2,64 3,47 2,39 4,81 3,86 5,82 3,50 6,55 2,65 3,38 2,68 4,51 3,84 5, 28 3,32 6,70 2,45 3,22 2,51 4,96 3,81 5,17 3,08 6,75 2,50 3,18 2,48 4,76 3,22 5,06 3,35 6,72 2,63 3,58 2,70 4,69 3,68 5,78 3,01 6,07 8,39 10,23 9,01 12,16 10,08 13,10 11,95 14,20 12 1,83 3,04 2,35 3,67 2,88 4,91 3,85 5,83 1,51 3,24 2,15 3,94 2,23 4,67 3,10 5,79 1,69 2,84 2,28 3,84 2,15 4,92 3,15 5,63 1,81 2,83 2,94 3,83 2,77 4,43 3,24 5,70 1,35 3,07 2,38 3,03 2,73 4,60 3,75 5,77 7,39 12,65 9,92 15,47 11,27 19,69 14,45 26,77 13 2,32 3,73 2,08 4,77 3,075 5,83 3,78 6,98 2,67 4,48 3,34 5,58 3,78 6,13 4,98 9,78 3,73 5,91 3,84 7,52 4,43 9,78 6,04 14,70 2,14 3,24 2,645 4,54 3,74 5,76 3,28 6,08 14 2,87 4,13 3,59 5,85 3,08 6,22 4,49 9,89 15 3,33 5,86 3,32 7,65 4,40 9,94 6,70 14,69 2,60 3,77 2,57 4,11 3,90 5,36 3,05 6,87 2,55 4,55 3,16 5,90 3,20 6,50 4,50 9,31 3,62 5,21 3,29 7,79 4,83 9,57 6,41 14,34 2,46 3,27 2,45 4,88 3,99 5,98 3,48 6,40 2,527 4,44 3,40 5,13 3,14 6,75 4,47 9,11 3,65 5,56 3,14 7,10 4,75 9,59 6,34 14,54 2,20 3,85 2,57 4,65 3,96 5,40 3,04 6,04 3,59 4,28 4,43 5,76 4,61 5,64 5,27 6,12 2,83 4,69 3,00 5,69 3,42 6,70 4,68 9,86 2,82 3,54 3,23 3,63 3,44 4,25 3,10 4,53 3,29 5,98 3,99 7,43 4,99 9,21 6,39 14,74 136 8,41 12,98 10,80 15,99 11,14 17,36 14,15 22,75 17 7,04 10,50 9,68 11,21 9,06 12,40 11,97 14,76 18 7,03 12,56 9,14 15,11 11,83 19,40 14,08 26,30 19 3,09 4,80 4,53 5,32 4,78 5,87 5,36 6,13 20 3,04 3,61 3,83 3,70 3,52 4,47 3,77 4,47 8,42 12,78 10,88 15,69 11,59 17,81 14,62 22,99 8,47 8,44 12,18 12,08 10,79 10,37 15,04 15,86 11,57 11,84 17,09 17,63 14,54 14,173 22,79 22,569 8,40 8,73 7,16 10,78 10,73 10,31 9,34 9,49 9,06 12,06 11,44 11,98 9,98 10,97 10,73 12,95 13,047 13,16 11,13 11,61 11,254 14,86 14,75 13,52 7,80 7,19 12,04 12,037 10,72 9,10 15,71 15,55 11,94 11,21 19,31 18,38 14,94 14,86 26,07 26,37 7,50 12,09 9,76 15,91 11,74 19,42 14,49 26,81 3,45 4,74 4,42 5,14 4,77 5,67 5,23 6,63 3,68 4,84 4,30 5,46 4,10 5,61 5,68 6,98 3,40 4,45 4,75 5,82 4,58 5,90 5,21 6,75 2,84 3,51 3,53 3,84 3,39 4,05 3,55 4,72 2,88 3,17 3,99 3,64 3,28 4,53 3,80 4,50 2,96 3,50 3,29 3,04 3,24 4,52 3,59 4,13 . . . . , . . 30.05.2017. 60×84 1/16. . . . 7,0. 2017 . , . . .- . . 7,63. 100 . , 137 . , 15