2.1.1
, , , - , . () , () : , . .
. , ( ). , . . () , , . , 10 .
, . . , , . , .
( ) ( ) . .
, . . .
(). , , . . . .
2.1.2
. .
, . () , , xi ( ) X, ni, i - . .
|
|
:
1) (xmin) (xmax) ;
2) () ( ), xmin xmax, ;
3) xi ni ;
4) ( ) n :
n = (2.1)
m . (2.1) , , .
. 1 ( ).
m , , .
, :
- X ;
- , xmin , xmax . , , .
. 2 ( ).
2.1.3
, () ni / n, . [. (2.1)], :
. (2.2)
( ): () . , , , xi ni / n (. 17). xi , ni / n (. .18).
, xi (. 20, 21).
2.1.4
F (x) [. (1.19)], . F *(x), ( ) pi ni / n, n → pi X = xi ( ). F *(x) F(x) X [. . 1.2.3, (1.20) (1.21)]:
F *(x) = (2.3)
|
|
F *(x)= (2.4)
F *(x) [ (2.4)] ( 1 2). , F (x) (. 3).
, , . , , ( 5 22). , .
2.1.5
2.1.5.1
:
= (2.5)
xi , n . n xi , . . ( ) .
, xi, :
xi () m, ni , n .
, xi (2.6) .
2.1.5.2
2. 2 :
2 = 2 , (2.8)
n ;
2 = 2 , (2.9)
m xi, ni ().
2.1.5.3
, , :
(2.10)
2.1.5.4
() , , . , :
(2.11)
x 0 , , , k , ni - , .
2.1.5.5
. , n , n - .
, :
= (2.12)
x 0 , , ; k ; n ; Ti -1 , ; ni .
|
|
:
();
( );
;
;
:
¾ ;
¾ ;
¾ S,
¾ M ;
¾ .
2.1
A, xi xi (ni) . . (. 1).
1. A
xi | ni | ni / n | |
xmin =0 | 4 | 0,0506 | 0,0506 |
1 | 13 | 0,1646 | 0,2152 |
2 | 14 | 0,1772 | 0,3924 |
3 | 24 | 0,3038 | 0,6962 |
4 | 16 | 0,2025 | 0,8987 |
5 | 3 | 0,0380 | 0,9367 |
6 | 3 | 0,0380 | 0,9747 |
xmax =7 | 2 | 0,0253 | 1,0000 |
n =79 | 1,0000 | - |
. x 0 7 ni / n 0 0,3038 ( 17 18).
17. 18. ࠠ A A
xi, , xi .
F *(x) , (2.4) , 1. :
F *(x) =
F *(x) 0 1 (. 19).
19. A
(2.6 2.9) (. 1).
(2.6)
(2.9)
2 = 2,3668.
( ) , , 2 () () EX ( ), DX ( ) , = 3 (. . 2.1.5.4). : = 3 (. . 2.1.5.5).
2.2
B, ni . . (. 2). :
2. B
ni | ni/n | ||
59-63 | 3 | 0,015 | 0,015 |
63-67 | 23 | 0,115 | 0,130 |
67-71 | 67 | 0,335 | 0,465 |
71-75 | 76 | 0,380 | 0,845 |
75-79 | 27 | 0,135 | 0,980 |
79-83 | 4 | 0,020 | 1,000 |
n = 200 | 1,000 | - |
x 61 81 ni / n 0.015 0,380 ( 20,21).
20. () B
21. B
, . (. 2), , F (x) - .22.
|
|
22. F (x) B
(2.6)
(2.9)
2 = 14,6176.
=
(2.11):
= 69 + 4 * .
(2.12):
= 71 + 4 *
2.2.1
, (). . , , , , .
, X, q, . . . , q = , :
1) . : - ;
2) - ;
3) , μ;
4) Me , μ ( );
5) .
, , () .
2.2.2
θ:
, ,
μ .
2 (. (2.8)) , :
E ,
() n → ∞ . , n . , , ( ):
, (2.13)
(n -1)= f ( n , (2.5)).
2.1.5.2 [. (2.8) (2.9)],
; (2.14)
. (2.15)
2.3
C ( 3), , σ2 σ :
(2.5), S 2 (2.14) S S = . 3.
3. C
xi | xi - | (xi - )2 |
-35 | -5,8 | 33,64 |
-32 | -2,8 | 7,84 |
-26 | 3,2 | 10,24 |
-35 | -5,8 | 33,64 |
-30 | -0,8 | 0,64 |
-17 | 12,2 | 148,84 |
-175 | - | 234,84 |
:
;
S2 = = 46,968;
S =
2.2.3
θ, , , , . , - , θ. | | ( ), .
|
|
() γ:
P (| |≤ ∆)= γ (2.16)
,
P ( ≤ ∆)= γ
P ( ≤ ≤ ∆)= γ (2.17)
(2.17) , | , ∆| (.. ) (, ) (2
, , , 0,90; 0,95; 0,98; 0,99 . . ( 1) α =1-γ .
γ, ∆ n . , .
2.2.4 μ σ
X ∈ N (μ, σ ), σ . μ . , , , - . , : ∈ N (μ, σ), i = 1, 2, , n. :
∈ N (μ, ).
(2.16)
P (| . (2.18)
C , (1.45) X , σ σ/ ε ∆ ( , ), :
P (| ) 1 = 2( ) 1 = γ, (2.19)
= .
∆ = . (2.20)
(2.17) (2.18), :
P (
(2 ∆).
. (2.19)
(
, (. 3) ( ).
2.2.5 μ σ
X ∈ N (μ, σ), σ , N (μ, σ) . ,
t =
S (. 2.14) , n , ࠠ (t -) f = n 1.
,
P (|
= t, .
( - ) t - (. 5). t >
P (|
, γ ≫ α, , 0,95≫0,05.
t - f = n 1.
(2.24),
P ( ) = γ, (2.25)
. (2.20). ( )
∆ = . (2.26)
2.2.6
, X ∈ N (μ, σ).
(2.27)
( ) f = n - 1. . - (. 4) χα 2, : P (χ2>χα 2) = α = 1- γ, α (. .2.3 . 23).
,
P ( . (2.28)
() . , ( ) ( 24):
P ( . (2.29)
23. 24.
( 4), (2.29), .
P ( . (2.30)
(2.28) (2.27) ,
) = γ, (2.31)
.
, , () σ:
≤ σ ≤ . (2.32)
2.4
, σ2 σ γ = 0,95, C ( 3 2.3).
γ = 0, 95 (2.25) (2.25), σ2 (2.31) σ (2.32), C (n =6).
( 5) = 2,571 ( f = 6-1 = 5; ,
( 4):
u 1 = 0, 831 u 2 = 12,83, p = .
, = -29,2; S 2 = 46,968; S = 6,85; n =6 (. 2.3):
- (2.26) ∆ = 2,571 * , :
- : (-29,2 - 7,19) ≤ ≤ (-29,2 + 7,19), -36,39 ≤ ≤ -22,01;
- : = ,
.. σ:
18,30
2.2.7 n
n. : , ? . ( ). , γ.
, , (2.20),
n =
, n () ( γ)
γ ( n. , , n. , (2.33) σ.
, σ n (2.26):
n =
(2.33), σ, n . (2.34) n , , .
, (2.33) (2.34) γ, n, () ∆.
2.5
C, 2.3, , ( ) n ∆ = 3 γ = 0,99.
(2.34) .
= 0,99, ∆ = 3 n = 6 : f = n -1 = 6 1 = 5; S = 6, 85;
t 0,99 = 4,03 ( ).
n = (, ),
f = n -1 = 85-1= 84, t 0,99 = 2,64.
:
n ′ = (2,64∙6.85/3)² = 36,33 ≈ 37, n ′ t 0,99 = 2,72
n ″=( 2,72∙6.85/3)² = 38,57 ≈ 39 .
2.3.1
, . , , μ = 5 μ ≠ 5 .. , .
θ ; , θ ( , θ = , θ () (, θ< .
() , , , .
, . - . . , , , , . , .
, , . () , ( , , ). :
¾ , 0 ( ), ( ), , ;
¾ , 0 () 1, (. . 25), , 𝛼.
, . a 𝛼 = 1 - 𝛾. 𝛼 (0,1, , 0,001). (b).
, α β ( ). , 0.
, α 1. - 0 q :
0: q = q 0.
1 :
1: q < q 0, 1: q > q 0 1: q ¹ q 0.
, ( 25). ( ), (), γ ( α = 1- γ).
:
1) () 0 1;
2) () α;
3) ( , α 1);
4) ( );
5) ;
6) (): ( . . 25 ), - 0 H 1 ( ), , 0 1 ( ).
α α α/2 α/2
k k k1 k2
) |
) |
) |
25. :
, ,
H 1 4, . , - 0: 𝜇 ≠ 5 = 7,2, .. > 𝜇, 1: 𝜇 > 5 1: 𝜇 = 5.
: 1, , 0, , 0 α . 0 . , 0: 𝜇 = 5, , , α, , , 0: 𝜇 = 5,5 0:
𝜇 = 4 .. , , . , , 0, () .
2.3.2 σ
, X ∈ N (μ, σ), σ . a - 0: ( , 0 = 5). 1: , 1: 1: . ( ) ()
, (2.35)
- ,
0 Z ∈ N (0, 1) (. . 1.2.8, 1.43 1.44).
(. 3) () ( x >0).
1: , , ( 26) :
P (Z < - (2.36)
- N - ( 3 = z). , - ( z) .
(Z >0), , [. (1.17)],
( ) = 1 . (2.37)
, ( - ) Z,
Z <- (2.38)
f (z) f (z)
- 0 z 0