, (2.3) , y (2.3). . , , , (2.3), . . . , , , (2.3), .
. , {xj} .
x 1, x 2,, xn , n > 1. . , x = xi yi 1, yi 2,, yi . ,
, i = 1,2,, n. (4.20)
si 2 = , i = 1,2,, n. (4.21)
( σ 2), ( , s 2 ),
si 2 Îc2(mi 1), (4.22)
( (4.22) c2- mi 1 .) ,
/ σ 2 Îc2(N n), (4.23)
N = m 1 + + m n. ( 4.2.2),
S 22 = (4.24)
σ 2, (2.3).
S 21 = , (4.25)
4.3.1. . (2.3) S 21 S 22 , σ 2.
ij, - (. 4.3.2)
= + . (4.26)
4.2.1 , (4.26), σ 2, c2 N 2 , N = m 1 + + mn. (4.26) Z2 Z1 . , Z2/ σ 2 c2 N n . 4.3.3, , Z1 / σ 2 c2 N 2 (N n) = n 2 , Z1/ σ 2 Z2 / σ 2 . , S 21 = Z1/(n 2) σ 2. , S 21 = Z1 σ 2/(σ 2(n 2)) S 22 = Z2 σ 2/(σ 2(N n)) .
4.3.2. . (4.26) , , (3.2).
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4.3.3. . x = (x 1,, xn) N (m, In), m = (m 1,, mn), Qi, i = 1,2 x 1 m 1,, xn mn. Qi Îc2(ri), Q 1 Q 2 ³ 0, Q 1 Q 2 Q 2 c2(r 1 r 2) c2(r 2) .
4.3.4. .
S 21 / S 22 Î F (n 2, N n), (4.27)
F Գ. , S 21 S 22 σ 2. , (4.27) (F -) F (n 2, N n) .
4.3.5. . . 4.2.8 (2.3).
4.3.6. . ( , β 1 = 0) , F - Գ. , , tn, Գ F 1, n. . , F 1, n T (b 1) b 10 = 0 (. 4.3.8). , -, , F 1, n (β 1¹0, β 1>0, β 1<0) (u, +¥). (ҳ u F 1, n 1 α, 1 α / 2, α ). -, , , , . , - , F - .
4.3.7. . , F , (, (3.3)) Գ F 1, n 2.
. (3.3), σ 2, c2(n 1). 4.2.1 , σ 2, c2(n 2). 4.3.3 (3.2), σ 2, c2(1) .
F = (4.27)
F 1, n 2.
4.3.8. . b 1,
F = .
T (b 1) β 1=0 ( (4.19)). , F - (4.27) β 1.
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9
. ( )
| |||
(1) | y(1) | ||
(2) | y(2) | ||
(m) | y(m) |
ճ
STATGRAPHICS
1. STATGRAPHICS, , . ( File→Save→Data File). .
:
X | 0,1 | 0,2 | 0,3 | 0,4 | 0,5 | 0,6 | 0,7 | 0,8 | 0,9 | 1,0 | 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 | 1,7 | 1,8 | 1,9 | 2,0 |
y | -0,02 | 0,44 | 0,51 | 0,67 | 0,69 | 1.04 | 1,14 | 1,37 | 1,77 | 2,12 | 2,47 | 2,9 | 3,5 | 3,99 | 4,06 | 4,54 | 4,99 | 5,36 | 5,99 |
2. Relate → Simple Regression. . () (). ϳ . ³ . . , , . . :
Regression Analysis - Linear model: Y = a + b*X
-----------------------------------------------------------------------------
Dependent variable: y
Independent variable: x
-----------------------------------------------------------------------------
Standard T
Parameter Estimate Error Statistic P-Value
-----------------------------------------------------------------------------
Intercept 0,264386 0,0422984 6,25048 0,0000
Slope 0,317228 0,0138497 22,905 0,0000
-----------------------------------------------------------------------------
Analysis of Variance
-----------------------------------------------------------------------------
Source Sum of Squares Df Mean Square F-Ratio P-Value
-----------------------------------------------------------------------------
Model 6,42941 1 6,42941 524,64 0,0000
Residual 0,220589 18 0,012255
-----------------------------------------------------------------------------
Total (Corr.) 6,65 19
Correlation Coefficient = 0,983274
R-squared = 96,6829 percent
Standard Error of Est. = 0,110702
The StatAdvisor
---------------
The output shows the results of fitting a linear model to describe
the relationship between y and x. The equation of the fitted model is
y = 0,264386 + 0,317228*x
Since the P-value in the ANOVA table is less than 0.01, there is a
statistically significant relationship between y and x at the 99%
confidence level.
The R-Squared statistic indicates that the model as fitted explains
96,6829% of the variability in y. The correlation coefficient equals
0,983274, indicating a relatively strong relationship between the
variables. The standard error of the estimate shows the standard
deviation of the residuals to be 0,110702. This value can be used to
construct prediction limits for new observations by selecting the
Forecasts option from the text menu.
3. Graphical Options. :
- plot of fitted model;
- residual versus x.
: , .
.
4. Comparison of Alternative Models ( ). :
Comparison of Alternative Models
--------------------------------------------------
Model Correlation R-Squared
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Linear 0,9833 96,68%
Square root-Y 0,9431 88,95%
Exponential 0,8637 74,59%
Reciprocal-Y <no fit>
Reciprocal-X <no fit>
Double reciproc <no fit>
Logarithmic-X <no fit>
Multiplicative <no fit>
Square root-X <no fit>
S-curve <no fit>
Logistic <no fit>
Log probit <no fit>
--------------------------------------------------
, STATGRAPHICS, , , . . .
5*. ( ..). 1 - 3 , , , , .
10
. ( ) y = a0 + 1x + 22 y = a0 + 1x + 22 + 33. .
ճ
1. STATGRAPHICS, . 9 .
2. Relate → Polynomial Regression. 䳿 . 9. 2- 3- . .
3. 2- 3- : . .
11