3.1
ma = ∑(xj − x0)a nj / ∑n, (3.1)
ma − a;
xj − ;
xo − , ;
nj − ;
∑n − .
k = xj − xo / i (3.2)
3.1
xj, | nj, | kj | njkj | njkj2 | njkj3 | njkj4 |
9,0 | -7 | -7 | -343 | |||
11,0 | -6 | -54 | -1944 | |||
13,0 | -5 | -50 | -1250 | |||
15,0 | -4 | -24 | -384 | |||
17,0 | -3 | -42 | -378 | |||
19,0 | -2 | -38 | -152 | |||
21,0 | -1 | -28 | -28 | |||
23,0 | ||||||
25,0 |
3.1
27,0 | ||||||
29,0 | ||||||
31,0 | ||||||
33,0 | ||||||
35,0 | ||||||
37,0 | ||||||
− | -35 | -909 |
m1 = ∑njkj / ∑n = -35 / 200 = -0,175 (3.3)
m2 = ∑njkj2 / ∑n = 1725 / 200 = 8,625 (3.4)
m3 = ∑njkj3 / ∑n = -909 / 200 = -4,545 (3.5)
m4 = ∑njkj4 / ∑n = 42069 / 200 = 210,345 (3.6)
3.2
= x o im1, (3.7)
xo − , ;
i − ;
m1 − .
M = 23,0 − 2 ∙ 0,175 = 22,65 ≈ 22,7 .
σ = i √m2 − m1² (3.8)
σ = 2 √8,625 − (-0,175)2 = 5,86 .
σ= √m2 − m1² (3.9)
σ = √8,625 − (-0,175)2 = 2,93
3.3
2 = m2 − m1²; (3.10)
μ3 = m3 −3m2m1 + 2m1³; (3.11)
4 = m4 − 4m3m1 + 6m2m12 − 3m14 (3.12)
μ2, μ3, μ4 − , ;
m1, m2, m3, m4 − , , .
μ2 = 8,625 − (-0,175)² = 8,594
μ3 = -4,545 − 3 ∙ 8,625 ∙ (-0,175) + 2(-0,175)³ = -0,027
4 = 210,345 − 4 ∙ (-4,545)(-0,175) + 6 ∙ 8,625 (-0,175)2 − 3(-0,175)4 = 208,745
3.4
ra = a / (√2)a (3.13)
r3 = 3 / (√2)3 = -0,027/ (√8,594)3 = -0,001
r4 = 4 / (√2)4 = 208,745 / (8,594)2 = 2,826
3.5
α1 = r3 = -0,001
|
|
.
mα ≈ √(6/N)
mα = √(6/200) = 0,173
tα = α1 / mα = 0,001 / 0,173 = 0,006
, .
j = r4 − 3 (3.14)
j = 2,826 − 3 = -0,174
.
mj ≈ 2√(6/N)
m j= 2√(6/200) = 0,346
tj = j / mj = 0,174 / 0,346 = 0,51
. .
3.2
yj, | nj, | kj | njkj | njkj2 | njkj3 | njkj4 |
8,0 | -7 | -7 | -343 | |||
10,0 | -6 | -6 | -216 | |||
12,0 | -5 | -20 | -500 | |||
14,0 | -4 | -24 | -384 | |||
16,0 | -3 | -30 | -270 | |||
18,0 | -2 | -44 | -176 | |||
20,0 | -1 | -32 | -32 | |||
22,0 | ||||||
24,0 |
3.2
26,0 | ||||||
28,0 | ||||||
30,0 | ||||||
− | -49 | -1519 |
m1 = ∑njkj / ∑n = -49 / 200 = -0,245
m2 = ∑njkj2 / ∑n = 687 / 200 = 3,435
m3 = ∑njkj3 / ∑n = -1519 / 200 = -7,595
m4 = ∑njkj4 / ∑n = 9891 / 200 = 49,455
M = 22 − 2 ∙ 0,245 = 21,51 ≈ 21,5 .
σ = 2 √3,435 − (-0,245)2 = 3,67 .
σ = √3,435 − (-0,245)2 = 1,84
2 = 3,435 − (-0,245)2 = 3,375
3 = -7,595 − 3 ∙ 3,345 ∙ (-0,245) + 2(-0,245)3 = -5,099
4 = 49,455 − 4 ∙ (-7,595)(-0,245) + 6 ∙ 3,435 ∙ (-0,245)2 − 3(-0,245)4 = 43,139
r3 = 3 / (√2)3 = -5,099 / (√3,375)3 = -0,820
r4 = 4 / (√2)4 = 43,139 / (3,375)2 = 3,787
α1 = r3 = -0,820
.
mα = √(6/200) = 0,173
tα = α1 / mα = 0,820 / 0,173 = 4,739
∑ j = 3,787 − 3 = 0,787
.
mj = 2√(6/200) = 0,346
tj = j / mj = 0,787 / 0,346 = 2,28
. .
4.1 ( )
xj, | xj- M | (xj- M)2 | xj, | xj- M | (xj- M)2 | ||
21,8 | -2,0 | 4,00 | 13,0 | -10,8 | 116,64 | ||
28,5 | 4,7 | 22,09 | 29,5 | 5,7 | 32,49 | ||
22,3 | -1,5 | 2,25 | 22,5 | -1,3 | 1,69 | ||
18,0 | -5,8 | 33,64 | 19,5 | -4,3 | 18,49 | ||
23,0 | -0,8 | 0,64 | 28,0 | 4,2 | 17,64 | ||
23,5 | -0,3 | 0,09 | 20,0 | -3,8 | 14,44 | ||
36,0 | 12,2 | 148,84 | 30,0 | 6,2 | 38,44 | ||
19,0 | -4,8 | 23,04 | 22,5 | -1,3 | 1,69 | ||
21,5 | -2,3 | 5,29 | 27,0 | 3,2 | 10,24 | ||
22,0 | -1,8 | 3,24 | 28,0 | 4,2 | 17,64 | ||
475,6 | ─ | 512,52 |
|
|
M = ∑x j / ∑n (4.1)
M = 475,6 / 20 ≈ 23,8 .
σ = √ ∑(xj − M)2 / ∑n − 1 (4.2)
σ = √ 512,52 / 20 − 1 = 5,19 .
mM = σ / √ ∑n
mM = 5,19 / √20 ≈ 1,16 .
M mM = 23,8 1,16 .
C = σ 100 / M
C = 5,19 ∙ 100 / 23,8 = 21,8 %
p = mM ∙ 100 / M
p = 1,16 ∙ 100 / 23,8 = 4,9 %
t1 = M / mM
t1 = 23,8 / 1,16 = 21
.
4.2 ( )
xj, | xj- M | (xj- M)2 | |
28,5 | 5,4 | 29,16 | |
18,0 | -5,1 | 26,01 | |
23,5 | 0,4 | 0,16 | |
19,0 | -4,1 | 16,81 | |
22,0 | -1,1 | 1,21 | |
29,5 | 6,4 | 40,96 | |
19,5 | -3,6 | 12,96 | |
20,0 | -3,1 | 9,61 | |
22,5 | -0,6 | 0,36 |
4.2
28,0 | 4,9 | 24,01 | |
230,5 | − | 161,61 |
M = 230,5 / 10 = 23,05 ≈ 23,1 .
σ = √ 161,61 / 10 − 1 = 4,24 .
mM = 4,24 / √10 ≈ 1,34 .
M mM = 23,1 1,34 .
C = 4,24 ∙ 100 / 23,1 = 18,4 %
p = 1,34 ∙ 100 / 23,1 = 5,8 %
t1 = 23,1 / 1,34 = 17
.
t = (M1 − M2) / √(m12 + m22) ≥ tst, (4.3)
1, 2 − ;
m1, m2 − ;
tst − .
t = (23,8 − 23,1) / √(1,342 + 1,162) = 0,40
U = n1 + n2 − 2
U = 20 + 10 − 2 = 28
tst 2,1; 2,8; 3,7.
t < tst
.
.
F = σ12 / σ2²; (4.4)
σ1, σ2 − , σ1² σ2² .
F = 5,192 / 4,242 = 1,50
U1 = n1 − 1 = 19
U2 = n2 − 1 = 9
F 0,05 = 2,9
F < F 0,05
5%- .
4.3 ( )
yj, | yj- M | (yj- M)2 | yj, | yj- M | (yj- M)2 | ||
21,6 | -0,8 | 0,64 | 16,4 | -6,0 | 36,00 | ||
27,0 | 4,6 | 21,16 | 26,4 | 4,0 | 16,00 | ||
23,4 | 1,0 | 1,00 | 19,1 | -3,3 | 10,89 | ||
16,7 | -5,7 | 32,49 | 18,0 | -4,4 | 19,36 | ||
22,5 | 0,1 | 0,01 | 24,5 | 2,1 | 4,41 | ||
23,0 | 0,6 | 0,36 | 20,5 | -1,9 | 3,61 | ||
29,5 | 7,1 | 50,41 | 25,0 | 2,6 | 6,76 | ||
17,8 | -4,6 | 21,16 | 23,7 | 1,3 | 1,69 |
4.3
22,4 | 24,0 | 1,6 | 2,56 | ||||
23,8 | 1,4 | 1,96 | 23,1 | 0,7 | 0,49 | ||
448,4 | ─ | 230,96 |
M = 448,4 / 20 = 22,42 ≈ 22,4 .
σ = √ 230,96 / 20 − 1 = 3,49 .
mM = 3,49 / √20 ≈ 0,78 .
M mM = 22,4 0,78 .
C = 3,49 ∙ 100 / 22,4 = 15,6 %
p = 0,78 ∙ 100 / 22,4 = 3,5 %
t1 = 22,4 / 0,78 = 29
.
4.4 ( )
yj, | yj- M | (yj- M)2 | |
27,0 | 5,0 | 25,00 | |
16,7 | -5,3 | 28,09 | |
23,0 | 1,0 | 1,00 | |
17,8 | -4,2 | 17,64 | |
23,8 | 1,8 | 3,24 | |
26,4 | 4,4 | 19,36 | |
18,0 | -4,0 | 16,00 | |
20,5 | -1,5 | 2,25 | |
23,7 | 1,7 | 2,89 |
4.4
|
|
23,1 | 1,1 | 1,21 | |
220,0 | ─ | 116,59 |
M = 220 / 10 = 22,0 .
σ = √ 116,59 / 10 − 1 = 3,60 .
mM = 3,60 / √10 ≈ 1,14 .
M mM = 22,0 1,14 .
C = 3,60 ∙ 100 / 22,0 = 16,4 %
p = 1,14 ∙ 100 / 22,0 = 5,2 %
t1 = 22,0 / 1,14 = 19
.
t = (22,4 − 22,0) / √(0,782 + 1,142) = 0,29
U = 20 + 10 − 2 = 28
t < tst
.
.
F = 3,62 / 3,492 = 1,06
U1 = n1 − 1 = 19
U2 = n2 − 1 = 9
F 0,05 = 2,9
F < F 0,05
5%- .