SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Uncertaintyy evaluation on weighing designs M. Sc. Luis Omar Becerra
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Weighing designs Same nominal value, Example:
1 kg, 1 kg (*), 1 kg (**), 1 kg (***)
Subdivision for submultiples of the kilogram Example:
1 kg, 500 g, 200 g, 200 g (*), 100 g, 100 g (*)
S bdi i i for Subdivision f multiples l i l off the h kilogram kil Example:
1 kg, 1 kg (*), 2 kg, 2 kg (*), 5 kg, 10 kg
Luis O. Becerra2
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Example: 1 kg, 1 kg (*), 1 kg (**), 1 kg (***) 1 kg
1 kg (*)
‐ ‐ ‐
+
1 kg (**)
1 kg (***)
= y1
+
= y2
+ ‐ ‐
Observations
+ ‐
= y3 = y4
+ +
= y5 = y6
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Example: 1 kg Standard
‐ ‐
1 kg, 500 g, 200 g, 200 g (*), 100 g, 100 g (*) 500 g 200 g
+ + ‐ ‐
+ + + + ‐ ‐
200 g (*)
100 g
+ + + + +
+
‐
100 g (*)
Observations
= y1
+ +
= y2 = y3
+
= y4 = y5
+ + ‐
+ + +
= y6 = y7 = y8
4
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Example: 1 kg Standard
1 kg, 1 kg (*), 2 kg, 2 kg (*), 5 kg, 10 kg 1 kg (*)
‐ ‐ ‐ ‐ ‐ ‐ ‐
‐ ‐ +
2 kg
2 kg (*)
5 kg
10 kg
Observations
‐ ‐ ‐ ‐ ‐
‐ ‐ ‐ ‐ + +
‐ ‐ + +
+ +
= y1
+
= y2 = y3 = y4 = y5 = y6 = y7 = y8
5
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Subdivision methods •Lagrange operator
•Gauss Markov (Weighing Least Squares)
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Evaluation of the uncertainty for Lagrange O Operator t approach h Basic mass equation
⎛ ⎞ −1 ⎟ ⎜ − = − − m m ρ V V + Δ ISb −εd ∑j j ∑k k ∑ a⎜∑ k j⎟ j ⎝ k ⎠
Law of propagation of uncertainty for a univariate model
uc =
n
∑[c ⋅ u( x )]
2
i
i
i
=
⎡ ∂Y ⎤ ∑i ⎢ ∂X ⋅ u( xi )⎥ ⎣ i ⎦ n
2
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Lagrange operator Example: 1 kg, 1 kg (*), 1 kg (**), 1 kg (***) N Normal l equations ti
X T Xβ = X T Y
⎡ 3 − 1 − 1 − 1 1⎤ ⎡ β R ⎤ ⎡− 1 − 1 − 1 0 ⎢− 1 3 − 1 − 1 0⎥ ⎢ β ⎥ ⎢ 1 0 0 −1 ⎢ ⎥ ⎢ 1⎥ ⎢ ⎢− 1 − 1 3 − 1 0⎥ ⋅ ⎢ β 2 ⎥ = ⎢ 0 1 0 1 ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 0 ⎢− 1 − 1 − 1 3 0⎥ ⎢ β 2 ⎥ ⎢ 0 0 0 0 0 ⎦⎥ ⎢⎣ λ ⎥⎦ ⎢⎣ 0 0 0 0 ⎣⎢ 1
Solution
(
⎡ y1 ⎤ ⎢ ⎥ 0 0 0⎤ ⎢ y2 ⎥ − 1 0 0 ⎥⎥ ⎢ y3 ⎥ ⎢ ⎥ y ⎥ ⋅ 0 −1 0 ⎢ 4 ⎥ ⎥ ⎢ 1 1 0 ⎥ y5 ⎥ ⎢ ⎥ ⎥ y 0 0 1⎦ ⎢ 6 ⎥ ⎢m ⎥ ⎣ 1kg ⎦ −1 T
)
βˆ = X'T X' X' Y'
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Contributions of uncertainty, Ratio of mass between the unknown and the standard
M Mass standard t d d
Air density
Ciu(xi)
hj =
mj mk
, j = 1...k
h j u (mk ) u ( ρ a )(V j − h jVk ) Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Contributions of uncertainty
Volume of the standard
Volume of j
ρ a h j u (Vk ),
j = 1...k
ρ a u (V j )
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Contributions of uncertainty Difference of Indications
Sensitivity of the balance
Resolution of the balance
s (ΔI ) Sb u (ΔI ) = Sb n −1
⎛ u (ms ) u (I s ) ⎞ ⎟⎟ ΔIu (Sb ) = ΔI ⎜⎜ + ΔI s ⎠ ⎝ m2
d u (ε d ) = 2 12 Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Contributions of uncertainty Fitting of the Least squares
[
u j ( fit ) = s 2 ( X T X) j , j
−1
]
1
2
[ Y − Xβˆ ] [Y − Xβˆ ] = T
with
s
2
m−n
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Standard uncertainty
[
]
u 2 (m j ) = (h j u (mk ) ) + (u ( ρ a )(V j − h jVk )) + ρ a (u (V j ) + h j u (Vk )) 2
2
2
2 ⎡ ⎤ ⎛ ⎞ u (ms ) u (I s ) d ⎞ ⎛ s (ΔI ) ⎞ ⎛ 2 ⎟ ⎜ + + Δ + u +⎜ I 2 + ⎟ ⎢ ⎜ ⎟ ⎥ ⎜ j ( fit ) ⎟ ΔI s ⎠⎦ ⎝ 12 ⎠ ⎝ Sb n ⎠ ⎣ ⎝ m2 2
2
u(m j ) =
n
2 [ ] c ⋅ u ( x ) ∑ i i i
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Degrees of freedom
ν eff (m j ) =
u 4 (m j )
∑
[ciu (xi )]
4
νi
Expanded uncertainty
U (m j ) = t(ν ,α ) × u(m j ) Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Evaluation of the uncertainty for Gauss Markov approach h Basic mass equation
⎛ ⎞ −1 ⎟ ⎜ − = − − m m ρ V V + Δ ISb −εd ∑j j ∑k k ∑ a⎜∑ k j⎟ j ⎝ k ⎠ Law of propagation of uncertainty for a univariate model
uc =
n
∑[c ⋅ u( x )]
2
i
i
i
=
⎡ ∂Y ⎤ ∑i ⎢ ∂X ⋅ u( xi )⎥ ⎣ i ⎦ n
2
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Gauss Markov approach Example: 1 kg, 1 kg (*), 1 kg (**), 1 kg (***) 0 ⎡− 1 1 ⎢− 1 0 1 ⎢ ⎢− 1 0 0 ⎢ ⎢ 0 −1 1 ⎢ 0 −1 0 ⎢ 0 −1 ⎢0 ⎢1 0 0 ⎣
Solution
0⎤ ⎡ y1 ⎤ ⎡ e1 ⎤ ⎢ y ⎥ ⎢e ⎥ 0⎥⎥ 2 2 ⎡β R ⎤ ⎢ ⎥ ⎢ ⎥ 1 ⎥ ⎢ ⎥ ⎢ y3 ⎥ ⎢ e3 ⎥ ⎥ ⎢ β1 ⎥ ⎢ ⎥ ⎢ ⎥ = y − e 0⎥ ⋅ ⎢β2 ⎥ ⎢ 4 ⎥ ⎢ 4 ⎥ 1 ⎥ ⎢ ⎥ ⎢ y5 ⎥ ⎢ e5 ⎥ ⎥ ⎣ β3 ⎦ ⎢ ⎥ ⎢ ⎥ 1⎥ ⎢ y6 ⎥ ⎢e6 ⎥ ⎢m ⎥ ⎢ 0 ⎥ 0⎥⎦ ⎣ R⎦ ⎣ ⎦
(
−1 Y
βˆ = X Ψ X T
Xβ = Y − e
)
−1
X T Ψ −Y1 Y Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Variance – Covariance Matrix
[
Ψ Y = E (y − E(y ))(y − E(y ))
( )
E eeT
⎡ σ y21 E (e1e2 ) ⋅ ⎢ 2 σ y2 ⎢ ⎢ ⋅ = ΨY = ⎢ ⎢ ⎢ ⎢ ⎢⎣ symm
T
⋅
⋅
⋅ ⋅
]
E (e1en )⎤ ⎥ ⎥ ⋅ ⎥ ⎥ ⋅ ⎥ ⋅ ⎥ ⎥ 2 σ yn ⎥⎦ Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Construction of complete variance - covariance matrix of Y
n
Ψ Y = ∑ J xi Ψ xi J
T xi
i
Law of propagation of uncertainty for a multivariate model
analogous to
u ( y) = 2
⎡ ∂Y ⎤ ∑i ⎢ ∂X ⋅ u( xi )⎥ ⎣ i ⎦ n
2
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
With ⎡ ∂y1 ⎤ ⎢ ∂x ⎥ ⎢ i⎥ ⎢ ∂y2 ⎥ ⎢ ∂xi ⎥ ⎢ ⎥ ⎥ J xi = ⎢ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ∂yn ⎥ ⎢ ∂x ⎥ ⎣ i⎦
⎡σ 2 ( xi ) . . ⎢ 2 (xi ) . σ ⎢ ⎢ . ⎢ Ψ xi = ⎢ ⎢ ⎢ ⎢ ⎢ symm ⎣
.
.
.
.
.
0
.
.
.
.
.
.
.
.
σ 2 ( xi )
⎤ ⎥ . ⎥ . ⎥ ⎥ . ⎥ . ⎥ ⎥ . ⎥ σ 2 ( xi )⎥⎦ .
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Construction of complete variance - covariance matrix of Y,
ΨY
⎛ ⎞ −1 ⎜ ⎟ yi = ρ a i ⋅ ⎜ ∑ V j − ∑ Vk ⎟ + ΔI i Sbi − 2ε d k ⎝ j ⎠
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Contributions of matrix Ψ Mass standard
J ms Ψ ms J
Air density y
J ρa Ψ ρa J ρa
Volume of the Standard i
T ms
T
J Vi ΨVi J
T Vi
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Contributions of matrix Ψ Difference of Indications
S Sensitivity iti it off the th balance b l
Resolution of the balance
J ΔI i Ψ ΔI i J
T ΔI i
J Sbi Ψ Sbi J
T Sbi
Jε d Ψε d Jε d T
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Contributions of matrix Ψ Fitting of the
J fit Ψ fit J
Least squares
T fit
Solution of Gauss Markov Approach
(
−1 Y
βˆ = X Ψ X T
n
)
−1
X T Ψ −Y1Y
Ψ Y = ∑ J xi Ψ xi J
T xi
i
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
βˆ
Variance – Covariance Matrix of
(
−1 Y
cov = X Ψ X T
)
−1
element j,j of cov = u (m j ) 2
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Evaluation of the uncertainty by numerical simulation (Monte Carlo method)
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
JCGM 102 2009-06-11 Draft Evaluation of measurement data Supplement 2 to the “Guide to the expression of uncertainty in measurement" - Models with any number of output quantities
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
MCM Input quantities 3.25 coverage probability probability that the true quantity value is contained within a specified coverage interval or coverage region. 4.13 4 13 In the vector case, case the PDF for X is denoted by gX(ξ), where ξ = (ξ1,….., ξN)T is a variable describing the possible values of the quantity X. This X is considered as a random variable with expectation E(X) and covariance matrix V(X). Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
MCM Input quantities 4.4 Measurement function When the Yi are expressed directly as formulae in X, the measurementt model d l is i represented t d by b the th measurement function
Y = f (X ) Or equivalent
Y1 = f1 ( X ) . . Ym = f m ( X )
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
MCM Input quantities 7.2 Number of Monte Carlo trials 7.2.2 Because there is no guarantee that any specific pre-assigned number will suffice, a procedure that selects M adaptively, i.e. as the trials progress, can be used. used
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
MCM Propagation 7.3 Making draws from probability distributions 7.3.1.1 In an implementation of MCM, M vectors xr, r = ,1….,M, are drawn d f from th PDFs the PDF gXi (ξi) for f the th input quantities X1,….,XN. Draws would be made from the jjoint ((multivariate)) PDF gX(ξ) if appropriate. pp p Recommendations concerning the manner in which these draws can be obtained are given in GUM Supplement 1 for the commonest univariate distributions.
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
MCM Propagation 7.4 Evaluation of the model 7.4.1 The multivariate model is evaluated for each of the M draws from the PDFs for the N input quantities. Specially, denote the M draws by x1,….,xM, each of dimension N X 1, 1 where the rth th draw xr contains x1,r …. xNr, with xi,r a draw from the PDF for Xi. Then, the model values are
yr = f ( xr ),
r = 1,....,M Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Primary MCM Output: distribution function for the output quantity 7.5 Discrete representation of the distribution function for the output quantity A discrete representation p of the distribution function for the vector output quantity is formed from the M values of the vector output quantity obtained in 7.4. In general, this representation is a matrix G of dimension mxM whose rth column is the rth value of the vector output quantity. tit For F a univariate i i t model, d l G is i a row vector. t Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
MCM Summarizing 7.6 Estimate of the output quantity and the associated covariance matrix
1 ~ ( y1 + .... + yM ) y= M
[
1 ( y1 − ~y1 )( y1 − ~y1 )T + ... + ( yM − ~y1 )( yM − ~y1 )T U ~y = M −1
]
are taken, respectively, as an estimate y of Y and the covariance matrix Uy associated with y.
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
MCM Summarizing 7.7 Coverage region for a vector output quantity 7.7.1 The determination from G of coverage regions for vector output quantities is not straightforward, straightforward be-cause be cause the operation of sorting vector data is generally not welldefined. Some approaches have been proposed, including the sorting of such data using the metric
( yr − a )T Σ −1 ( yr − a ) where a is a location statistic, such as the average, for the set y1,…,yyM and Σ is a dispersion statistic, statistic such as the covariance matrix Uy, for the set. Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico Distribution for 5 kg/F89 X <=0 2.275% 2 275%
1.2
Valu ues in 10^ 3
1
Statistic
X <=0 97.725% 97 725% M ean = 3.051933E-03
0.8 0.6 04 0.4 0.2 0 -5
-4
-3
-2
-1
Values in 10^-3
Distribution for 5 kg/F89 X <=0 2.275%
Values in 10^ 3
1
X <=0 97.725% M ean = 3.051933E-03
0.8 0.6 0.4
0 -5
-4
-3
Values in 10^-3
-2
-1
Value
Minimum
-0.00473
5%
-0.00366
Maximum
-0.00116
10%
-0.00353
Mean
-0.00305
15%
-0.00343
Std Dev
0.00037
20%
-0.00336
Variance
1.36E-07
25%
-0.00330
Skewness
6.09E-03
30%
-0.00325
Kurtosis
3.00E+00
35%
-0.00319
Median
-0.00305
40%
-0.00314
Mode
-0.00339
45%
-0.00310
Left X
-0.00379
50%
-0.00305
Left P
2.28%
55%
-0.00301
Right X
-0.00231
60%
-0.00296
Right P
97 72% 97.72%
65%
-0.00291 0 00291
Diff X
0.00148
70%
-0.00286
Diff P
95.45%
75%
-0.00280
0
80%
-0.00274
Filter Min
85%
-0 00267 -0.00267
Filter Max
90%
-0.00258
95%
-0.00244
#Errors 0.2
Summary Statistics Value %tile
#Filtered
0
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Examples Weighing Desing (1, 1*, 1**, 1***) GM x u(x) 1 kg 0.430 mg 0.013 g g (*) () 0.563 mg 0.021 1 kg 1 kg (**) 0.542 mg 0.017 1 kg (***) 0.011 mg 0.016 0.0 χ2 evaluated 7.8 χ2 critical value
Weighing Desing (1, 1*, 1**, 1***) GM x u(x) g g 0.430 0 30 mg 0.013 0 0 3 1 kg 1 kg (*) 0.591 mg 0.032 1 kg (**) 0.556 mg 0.030 1 kg (***) 0.027 mg 0.030 12.1 χ2 evaluated critical t ca value a ue 7.8 8 χ2 c χ
mg g mg mg mg
GM-S II x 0.430 0.563 0.542 0.011
mg g mg mg mg
GM-S II x 0.430 0 30 0.591 0.556 0.027
LO mg g mg mg mg
u(x) 0.013 0.027 0.015 0.013
x mg 0.430 g mg 0.563 mg 0.542 mg 0.011 F evaluated F critical value
mg g mg mg mg
u(x) 0.013 0.027 0.015 0.013 0.3 3.9
mg g mg mg mg
LO-S II x 0.430 0.563 0.542 0.011
mg g mg mg mg
OL-S II x 0.430 0 30 0.588 0.554 0.023
OL mg g mg mg mg
u(x) 0.013 0 0 3 0.037 0.029 0.028
x mg g 0.430 0 30 mg 0.588 mg 0.554 mg 0.023 F evaluated Fc critical t ca value a ue
mg g mg mg mg
u(x) 0.013 0 0 3 0.041 0.034 0.033 1428.5 3.9 3 9
mg g mg mg mg
u(x) 0.013 0.027 0.015 0.013
mg g mg mg mg
mg g mg mg mg
u(x) 0.013 0 0 3 0.041 0.034 0.033
mg g mg mg mg
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Weighing Desing (10, 5, 2, 2*, 1, 1*) GM x u(x) 10 kg -8.43 mg 0.37 5 kg -3.03 mg 0.19 2 kg -0.96 mg 0.08 2k kg (*) -0.55 0 55 mg 0 08 0.08 1 kg 0.563 mg 0.027 1 kg (*) 0.016 mg 0.044 0.0 χ2 evaluated 8.0 χ2 critical value Weighing Desing (10, 5, 2, 2*, 1, 1*) GM x u(x) 10 kg -8.39 mg 0.60 5k kg -3.05 3 05 mg 0 31 0.31 2 kg -0.89 mg 0.13 2 kg (*) -0.54 mg 0.13 1 kg 0.563 mg 0.027 1 kg (*) -0.029 mg 0.084 2 evaluated l t d 23 7 23.7 χ2 8.0 χ2 critical value
mg mg mg mg mg mg
GM-S GM S II x -8.43 -3.03 -0.96 -0.55 0 55 0.563 0.016
mg mg mg mg mg mg
GM-S II x -8.40 -3.05 3 05 -0.89 -0.54 0.563 -0.029
LO mg mg mg mg mg mg
u(x) 0.55 0.28 0.11 0 11 0.11 0.027 0.055
x mg -8.43 mg -3.03 mg -0.96 mg -0.55 0 55 mg 0.563 mg 0.016 F evaluated F critical value
mg mg mg mg mg mg
u(x) 0.55 0.28 0.11 0 11 0.11 0.027 0.056 0.3 3.9
mg mg mg mg mg mg
LO-S LO S II x -8.43 -3.03 -0.96 -0.55 0 55 0.563 0.016
mg mg mg mg mg mg
OL-S II x -8.39 -3.05 3 05 -0.89 -0.54 0.563 -0.026
OL mg mg mg mg mg mg
u(x) 0.73 0 37 0.37 0.16 0.16 0.027 0.091
x mg -8.39 mg -3.05 3 05 mg -0.89 mg -0.54 mg 0.563 mg -0.027 F evaluated l t d F critical value
mg mg mg mg mg mg
u(x) 0.73 0 37 0.37 0.16 0.16 0.027 0.091 162 0 162.0 3.9
mg mg mg mg mg mg
u(x) 0.56 0.28 0.11 0 11 0.11 0.027 0.057
mg mg mg mg mg mg
mg mg mg mg mg mg
u(x) 0.73 0 37 0.37 0.16 0.16 0.027 0.092
mg mg mg mg mg mg
Luis O. Becerra
SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico
Weighing Desing (10, 5, 2, 2*, 1, 1*) GM x u(x) 1 kg -0.141 mg 0.010 500 g -0.045 mg 0.014 200 g -0.026 mg 0.010 200 g (*) 0 024 mg 0.024 0 010 0.010 100 g -0.036 mg 0.010 100 g (*) -0.163 mg 0.010 3.4 χ2 evaluated 11.1 χ2 critical value Weighing Desing (10, 5, 2, 2*, 1, 1*) GM x u(x) 1 kg -0.141 mg 0.010 500 g -0.061 0 061 mg 0 022 0.022 200 g -0.011 mg 0.016 200 g (*) 0.021 mg 0.016 100 g -0.036 mg 0.017 100 g (*) -0.156 mg 0.017 2 evaluated l t d 16 2 16.2 χ2 11.1 χ2 critical value
mg mg mg mg mg mg
GM-S GM S II x -0.141 -0.045 -0.026 0 024 0.024 -0.036 -0.163
mg mg mg mg mg mg
GM-S II x -0.141 -0.061 0 061 -0.011 0.021 -0.036 -0.156
LO mg mg mg mg mg mg
u(x) 0.009 0.014 0.011 0 011 0.011 0.010 0.010
x mg -0.141 mg -0.045 mg -0.026 mg 0 024 0.024 mg -0.036 mg -0.163 F evaluated F critical value
mg mg mg mg mg mg
u(x) 0.010 0.019 0.014 0 014 0.014 0.014 0.014 1.2 3.9
mg mg mg mg mg mg
LO-S LO S II x -0.141 -0.045 -0.026 0 024 0.024 -0.036 -0.163
mg mg mg mg mg mg
OL-S II x -0.141 -0.060 0 060 -0.011 0.021 -0.036 -0.156
OL mg mg mg mg mg mg
u(x) 0.009 0 022 0.022 0.016 0.016 0.017 0.017
x mg -0.141 mg -0.061 0 061 mg -0.011 mg 0.021 mg -0.036 mg -0.156 F evaluated l t d F critical value
mg mg mg mg mg mg
u(x) 0.010 0 027 0.027 0.020 0.020 0.020 0.020 60 6.0 3.9
mg mg mg mg mg mg
u(x) 0.010 0.015 0.011 0 011 0.011 0.011 0.011
mg mg mg mg mg mg
mg mg mg mg mg mg
u(x) 0.010 0 024 0.024 0.017 0.017 0.018 0.018
mg mg mg mg mg mg
Luis O. Becerra