. .
. .
:
;
, ;
;
, .
( ) (). (n). (xi). , xi, () (mi). ( ).
( ), ( ). . ; ( , ); , , . :
k = 1 + log2 n = 1 + 3,322× lg n. (1)
.
() , . : . .
, , . . . Q - , Q* , :
, .. ( , , );
, .. M (Q*) = Q;
, .. D (Q*) .
( ) ( )
|
|
, (2)
n , k . , ( ).
D
. (3)
(.. M (D) ¹ D), :
, (4)
, ( ) .
:
(5)
, . , .
. , () :
R = xmax xmin; (6)
()
; (7)
; (8)
; (9)
. (10)
( ) ( ).
, () g
, (11)
tg ( 1) F (tg) = 0,5 g, ( 2), f = n 1, a = 1 g.
. ( ) , , .
, , . ( ) . , . , , .
( ) . .
,
p(xi, y) , X xi, Y y. , :
|
|
Mx (Y) = f (x), My (X) = g (y).
( , ).
Mx (Y) = ryxx + b, My (X) = rxyy + d,
ryx rxy :
.
. , -1 £ r £ +1. r = 0. , , . |r| = 1.
, :
. (12)
,
- .
. (13)
. (14)
. . . .
() 0. () 1, . , , .
, . - . :
1. .
2. , .
3. .
4. , .
2 , .
, , . , , . :
, 0 ( ), ( );
, 0 () 1, .
0 q :
0: q = q 0.
1 :
1: q < q 0, 1: q > q 0 1: q ¹ q 0.
, . ( ) .
. a . , , (0,1, 0,05, 0,01, 0,001 ). (b), .. . .
:
|
|
1) 0 1;
2) ;
3) ;
4) ;
5) ( );
6) : , 0 H 1, , 0 1. , : 1, , 0, , . 0 .
.
I). . , , , , . a 0: . .
,
n 1 . () t. t. 1: a n 1. t < t, , . 1: . , | t| < t.
II) ( ). a 0: . , , , .
,
, M (Z) = 0, D (Z) = 1. z. z. 1: F (z) = 0,5 a. z < z, , . 1: F (z) = 0,5×(1 a). , | z| < z.
III) , ( ). a 0: .
,
(n + n 2) . t. t. (I).
IV) . a 0: D () = D (Y).
|
|
,
f 1 = n 1 f 2 = n 1 ( , n). () F. F 1: D () > D (Y) a f 1 f 2. , F < F.
V) . a 0: D ( 1) = D ( 2) = = D (l).
,
f = n 1 l (n , l ). , . .
VI) . a 0: r = 0. ( , ).
,
f = n 2 . (I).
VII) . n , m . , 0. a , 0. (.. , : ).
, . , 0, .
,
. , (I).
VIII) . . - , F (x). .
F* (x). 0 , . . c2 - .
, . . , ( 4), .
. . , . ( ).
, .
,
mi , mi , l . c2 l p 1 ( ). , .
, 0: ( 1) = ( 2) = ( l). . () ( , ). , .
|
|
, . : ( ) ( , ). . , . .
. ; , .
, (15)
. n , l , qi i - , xij , , i - . , . f1 = l 1, f2 = n l.
. . . , ( ).
:
(16)
A, B (), lA, lB A B, :
(17)
fA = lA 1, fB = lB 1, fAB = (lA 1)(lB 1), f2 = lAlB (q 1).
1. , , . ( ) :
17, 49, 79, 44, 82, 65, 109, 77, 107, 88, 62, 64, 79, 55, 44, 53, 140, 50, 67, 85, 64, 94, 41, 38, 62, 77, 154, 52, 56, 80, 49, 80, 79, 52, 65, 52, 127, 49, 73, 43, 95, 65, 83, 85, 111, 95, 112, 64, 94, 86, 34, 62, 101, 67, 59, 103, 67, 47, 65, 79, 64, 77, 32, 68, 145, 56, 172, 79, 67, 53, 35, 79, 70, 88, 137, 49, 125, 37, 65, 71, 35, 50, 37, 171, 139, 88, 137, 71, 77, 34, 62, 64, 79, 95, 124, 50, 127, 67, 67, 82, 14, 136, 76, 122, 82, 67, 111, 67, 70, 94, 73, 95, 94, 65, 80, 94, 160, 140, 95, 89, 65, 79, 80, 112, 35, 80, 109, 148, 127, 124, 68, 49, 70, 125, 88, 77, 119, 64, 148, 71.
n = 140 . , 10 ( ), 10 ( ). .
, . g = 0,95 .
. a = 0,05.
. , 14 172 . (1) :
k = 1 + 3,322 × lg n = 1 + 3,322×lg140 = 1 + 3,322×2,146 = 8,129.
( ): k = 9. :
.
20 ( 10 ). 10 ( xmin = 14 ).
. (, ). 1. ( ).
1
() xi | 10-30 | 30-50 | 50-70 | 70-90 | 90-110 | 110-130 | 130-150 | 150-170 | 170-190 |
mi | |||||||||
wi | 0,0143 | 0,1357 | 0,3000 | 0,2571 | 0,1071 | 0,0929 | 0,0643 | 0,0143 | 0,0143 |
F*(x) | 0,0143 | 0,1500 | 0,4500 | 0,7071 | 0,8143 | 0,9071 | 0,9714 | 0,9857 | 1,0000 |
(. 1) (. 2).
(2) :
( xi ).
(3) (5) :
.
.. (n = 140), .
(11)
,
tg ( 1) (tg) = 0,5 g = 0,5∙0,95 = 0,475. g = 0, 95 = (80,0 5,4) (74,6 < < 85,4) .
, :
H 0: ;
H 1: .
, (. 1), .. mi 4. 2 ( mi), ¥, +¥.
, 80,0 32,34 , 1 pi:
;
.
(n = 140) (mi).
.
2
, | mi | pi | mi | mi mi | |
(¥, 50) | 0,1762 | 24,67 | -3,67 | 0,55 | |
(50, 70) | 0,2021 | 28,29 | 13,71 | 6,64 | |
(70, 90) | 0,2434 | 34,08 | 1,92 | 0,11 | |
(90, 110) | 0,2021 | 28,29 | -13,29 | 6,25 | |
(110, 130) | 0,1156 | 16,18 | -3,18 | 0,63 | |
(130, 150) | 0,0452 | 6,33 | 2,67 | 1,13 | |
(150, +¥) | 0,0154 | 2,16 | 1,84 | 1,58 | |
S | 16,87 |
.
l = 7 . = 2 ( ). k = l - p - 1 = 7 - 2 - 1 = 4. a = 0,05 c2 ( 3).
.. , : .
2. - 6 . ( ):
6; 6; 3; 5; 4; 6; 6; 6; 6; 5; 6; 6; 4; 6; 4; 6; 5; 3; 4; 5; 3; 6; 4; 6; 6; 3; 4; 4; 5; 3; 4; 3; 4; 5; 4; 5; 6; 5; 5; 6; 5; 5; 6; 6; 6; 6; 5; 5; 6; 6; 4; 5; 5; 6; 6; 4; 5; 6; 6; 5; 4; 4; 4; 2; 4; 5; 4; 3; 4; 4; 5; 4; 5; 5; 5; 6; 6; 6; 4; 5; 4; 3; 5; 5; 6; 6; 4; 6; 4; 6; 5; 4; 5; 4; 5; 5; 4; 6; 5; 4; 4; 4; 5; 6; 5; 5; 2; 5; 6; 3; 6; 5; 4; 5; 6; 6; 4; 3; 5; 5; 4; 3; 2; 4; 4; 5; 5; 6; 5; 6; 6; 6; 5; 4; 6; 5; 4; 5; 5; 3; 5; 4; 6; 5; 5; 4; 4; 1; 6; 6; 5; 5; 4; 5; 5; 6; 3; 6; 6; 4.
n = 160 . .
, , , . g = 0,99 .
. a = 0,05.
. ( ) 1 6.
(, 1, 2, 3, 4, 5 6 ). 3. ( ).
3
xi | ||||||
mi | ||||||
wi | 0,00625 | 0,01875 | 0,08125 | 0,2625 | 0,325 | 0,30625 |
F*(x) | 0,00625 | 0,025 | 0,10625 | 0,36875 | 0,69375 |
(. 3).
(2) :
.
(3) (5) :
.
.. (n = 160), .
, ; Mo = 5 ( 52). ; Me = 5 ( 160 , ).
(11)
,
tg ( 1) (tg) = 0,5 g = 0,5∙0,99 = 0,495. g = 0,99 = (4,8 0,22) (4,58 < < 5,02).
, :
H 0: ;
H 1: .
, , , 0 (. 3), .. mi 4. 4 ( mi).
, 4,80. .. M (X) = np, . 0, 1, 2, 3, 4, 5 6 :
;
;
;
;
;
;
.
(n = 160) (mi).
.
4
xi | mi | pi | mi | mi mi | |
0, 1, 2 | 0,01696 | 2,7136 | 1,2864 | 0,609826 | |
0,081920 | 13,1072 | -0,1072 | 0,000877 | ||
0,245760 | 39,3216 | 2,6784 | 0,18244 | ||
0,393216 | 62,91456 | -10,9146 | 1,893483 | ||
0,262144 | 41,94304 | 7,05696 | 1,187341 | ||
S | 3,874 |
.
l = 7 . = 1 ( ). k = l - p - 1 = 7 - 1 - 1 = 5. a = 0,05 c2 ( 3).
.. , : , .
2. . - :
xi | |||||||||
mi |
, . a = 0,1.
. (2) :
.
, :
H 0: ;
H 1: .
, , .. mi 4. 5 ( mi).
, 2,46. :
; ;
; ;
; .
, 5
P (>5) = 1 (P (0) + P (1) + P (2) + P (3) + P (4) + P (5)) =
= 1 (0,0854 + 0,2102 + 0,2585 + 0,2120 + 0,1304 + 0,0641) = 0,0394.
(n = 160) (mi).
.
5
xi | mi | pi | mi | mi mi | |
0,0854 | 12,81 | -0,81 | 0,051218 | ||
0,2102 | 31,53 | -3,53 | 0,395208 | ||
0,2585 | 38,775 | 4,225 | 0,460364 | ||
0,2120 | 31,8 | 6,2 | 1,208805 | ||
0,1304 | 19,56 | -4,56 | 1,063067 | ||
0,0641 | 9,615 | -3,615 | 1,35915 | ||
>5 | 0,0394 | 5,91 | 2,09 | 0,739103 | |
S | 5,277 |
.
l = 7 . = 1 ( ). k = l - p - 1 = 7 - 1 - 1 = 5. a = 0,1 c2 ( 3).
.. , : , .
2. :
xi | |||||||
mi |
, . a = 0,1.
. :
H 0: ;
H 1: .
, . (k ). (n = 180) mi = 0,1429×180 = 25,714 ( ).
.
6
xi | mi | pi | mi | mi mi | |
0,1429 | 25,714 | 0,286 | 0,003181 | ||
0,1429 | 25,714 | -4,714 | 0,864191 | ||
0,1429 | 25,714 | -7,714 | 2,31414 | ||
0,1429 | 25,714 | 6,286 | 1,536665 | ||
0,1429 | 25,714 | 0,286 | 0,003181 | ||
0,1429 | 25,714 | 0,286 | 0,003181 | ||
0,1429 | 25,714 | 5,286 | 1,086637 | ||
S | 5,811 |
.
l = 7 . = 0 ( ). k = l - p - 1 = 7 - 0 - 1 = 6. a = 0,1 c2 ( 3).
.. , : , .
3. 8 . . : 902,4; 901,3; 898,4; 903,5; 901,1; 900,4; 899,7 900,3 (/). 7 . 905,5; 910,3; 903,8; 902,4; 899,9; 903,3 905,6 (/).
, . , () , , , .
, , ( ).
. ( ).
a = 0,05.
. (2-5) , :
, .
(6-5) :
R = xmax xmin = 903,5 898,4 = 5,1;
()
;
;
.
g = 0,95 ( a = 0,05) ( 2) f = 8 1 = 7 tg = 2,36.
.
g = 0,95 (900,76 1,24) / (899,52; 902,00) /.
:
, .
R = ymax ymin = 905,5 899,9 = 5,6;
; ;
.
g = 0,95 ( a = 0,05) ( 2) f = 7 1 = 6 tg = 2,45.
.
g = 0,95 (902,69 1,63) / (901,03; 904,32) /.
:
H 0: Dx = Dy;
H 1: Dx < Dy.
.
f 1 = n 1 = n 1 = 7 1 = 6 f 2 = n 1 = n 1 = 8 1 = 7 ( ). ( 4) a = 0,05 F = 3,87. .. F < F, (.. , ).
:
H 0: ;
H 1: .
f = n + n 2 = 8 + 7 2 = 13. ( 2) a = 0,05 t = 2,16. .. t > t, , .
4. . . :
7
x \ y | 0 20 | 20 40 | 40 60 | 60 80 | 80 100 |
0 20 | |||||
20 40 | |||||
40 60 | |||||
60 80 | |||||
80 100 |
x , y .
. . . , a = 0,01.
. 7 , ; , , ( 8).
8
x \ y | 0 20 | 20 40 | 40 60 | 60 80 | 80 100 | mx | |
0 20 | |||||||
20 40 | |||||||
40 60 | |||||||
60 80 | |||||||
80 100 |  
: 2017-02-28; !; : 1778 | : : - - , .
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: 0.24 .