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# Practical applications of sampling theory

DisciplinaProcessamento de Minerais I211 materiais2.063 seguidores
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```min in a Retsch Spectro
Mill. Mineral\u2013KBr mixture (200 mg) was pressed into a
tablet for the IR measurement. It was evaluated that the
size of the IR beam covered 38% of the area of the sample
tablet. The method was developed for quartz concen-
trations from 1% to 10%.
Dilution factor in 0.2 g/5.0 g = 0.004 is needed to
evaluate aL, the concentration of quartz in KBr tablets.
The procedure has three steps generating sampling errors.
These are
(1) taking the 20-mg mineral sample from the homogenized
mineral mixture to be mixed in KBr:
lot size =ML1 = 5 g
sample size =MS1 = 0.02 g
(2) the calibration tablet preparation:
lot size =ML2 = 5 g
sample size =MS2 = 0.2 g
(3) IR measurement:
lot size =ML3 = 200 mg
sample size =MS3 = 38% of 0.2 g = 76 mg
Following material properties were estimated:
d= 0.045 mm . . .particle size of quartz
g1 = 0.25 . . .estimated size distribution factor
aL= 1.0\u201310% . . .concentration of quartz mineral
a= 100% mixture before dilution with KBr
qc = 2.65 g/cm
3
qm= 3.0 g/cm
3 (in mineral mixture)
qm= 3.2 g/cm
3 (in KBr mixture)
f = 0.5
b = 1 . . .liberated particles
sampling
primary samples are taken. r1 is the standard deviation
between the means of the N1 strata, and c1 is the unit
cost of selecting a strata for sampling (usually c1 is
practically zero, because it only involves making the
decision from which strata the samples should be
taken).
4.1.2. Primary samples
From each selected stratum, n2 primary samples are
taken. N2 is the size of stratum expressed as the number
of potential samples that could be taken from the each
stratum. r2 is the standard deviation of primary samples (or
the within-strata standard deviation), and c2 is the unit cost
of taking a primary sample.
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u201394 89
Theory of stratified sampling can be used to optimize
sampling plans. This subject has been treated, e.g., by
Sommer  and Cochran . Sommer\u2019s approach is
followed in this presentation. Sometimes, stratification is
natural, e.g., if the lot to be investigated consists of bags,
Results of the fundamental sampling error estimation
are shown as function of the quartz in the original
sample mixture without dilution with KBr in Fig. 3, for
each step separately and for the total three-step calibration
procedure.
4. Optimization of sampling plans based on stratified
Fig. 3. Standard deviation estimates obtained in Example 3 for the three
sampling steps (1\u20133) and for the total standard deviation (4) in the
calibration of the IR spectrometer for quartz determination.
containers, wagon loads, etc. In sampling process streams,
where no clear stratum borders can be found, the strata
can be selected by the sampler. As Gy and Sommer 
have shown, stratified sampling usually gives smaller
uncertainties for the mean value, at worst equal to random
sampling.
4.1. Optimization of nested (hierarchical) sampling plans
for lots consisting of strata of equal sizes
Nested sampling plan is described in Fig. 4. In nested
sampling, the samples are taken at k levels (here k = 3). All
levels contribute to the overall uncertainty of the mean of
the lot, and at each level below the first sampling level, the
sample of the upper level is treated as the lot for this level in
the sampling chain. The quantities shown in Fig. 4 are
discussed in the following subsections.
4.1.1. Lot
The lot consists of N1 strata (sublots) of equal
sizes. Of these, n1 strata are selected from which the
4.1.3. Analytical samples
At this level, n3 is the number of analytical samples
prepared from each primary sample, N3 is the size of the
primary sample, as the number of potential analytical
samples that could be prepared from it, r3 is the standard
deviation of the preparation of the analytical samples (i.e.,
the standard deviation between analytical samples taken
from a primary sample), and c3 is the unit cost of preparing
an analytical sample.
For optimization purposes, the unit costs, ci, can be
given either as currency units or as relative costs, e.g., as
time required to carry out the given sampling operation.
Because the strata and the units within the strata are in
most cases autocorrelated, especially in process analysis,
and the sampling variances depend on sampling strategy
(systematic, stratified, or random selection), the variances
should in general be estimated by using Gy\u2019s vario-
graphic method. Analysis of variance based on the
design shown in Fig. 4 is recommended in many
statistical textbooks as the method for estimating the
variance components. Because this method does not take
autocorrelation into account, it should be used only in
case that strictly random sample selection is used (not
recommended), or there is no autocorrelation between the
sampling units.
Fig. 4. Nested sampling plan.
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u20139490
Because the strata have equal sizes, the mean of the lot
can be calculated as the unweighted mean of the analytical
results
Total number of samples analyzed : nt ¼ n1n2n3 ð6Þ
Mean of the lot : x¯ ¼ 1
n1n2n3
Xn1
i
Xn2
j
Xn3
k
xijk ð7Þ
Variance of the lot mean:
r2x¯ ¼
N1 \ufffd n1
N1 \ufffd 1
r21
n1
þ N2 \ufffd n2
N2 \ufffd 1
r22
n1n2
þ N3 \ufffd n3
N3 \ufffd 1
r23
n1n2n3
ð8aÞ
Eq. (8a) shows that if a sample can be taken from
every stratum (N1 = n1), the between-strata variance is
completely eliminated from the variance of the mean.
On the other hand, if at all levels, the sample is small
in comparison to the lot, from which it is taken, this
equation simplifies to
r2x¯ ¼
r21
n1
þ r
2
2
n1n2
þ r
2
3
n1n2n3
if all nibNi ð8bÞ
Total cost of the investigation : ct ¼ n1c1 þ n1n2c2
þ n1n2n3c3 ð9Þ
This system can be optimized in two ways: either so
that the maximum tolerable variance is first specified and
the total cost has to be minimized, or the total cost is
fixed and the variance of the mean has to be minimized.
An exact mathematical solution for this optimization
problem cannot be derived, because the number of sam-
ples taken can only be an integer number. The optimum
can be found however, either by checking all feasible
solutions, which is relatively easy by the speed of modern
computers, or by using approximate mathematical solution
given by Sommer . Approximate mathematical solution
can be derived by assuming that all ni are continuous
instead of integers and all NiHni. The mathematical
solution is presented below.
4.1.3.1. Maximum costs, cmax fixed, variance of the mean
minimized. For the levels below, the first level the number
of samples are to be taken can be evaluated by using the
formula
ni ¼ si
si\ufffd1
\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03
ci\ufffd1
ci
r
; constrained to integers 1VniVNi;
for levels i > 1 ð10Þ
By substituting the values for ni (i>1) in Eq. (9), n1 can
now be solved
n1 ¼ cmax
c1 þ n2c2 þ n2n3c3 ; rounded to the lower integer
1Vn1VN1 ð11Þ
4.1.3.2. Target value for the variance of the mean fixed,
total cost minimized. If rT
2 is the target value of the
variance of the mean of the procedure, then the protocol
has to be designed so that the variance of the mean of the
selected procedure, rx¯
2V rT
2.
For levels i >1, Eq. (10) also applies in this case. Number
of sublots to be sampled at level 1 can now be solved either
from Eq. (8a) (exact solution, Eq. (12a)) or from Eq. (8b)
(approximate solution, Eq. (12b), when all NiHni)
n1¼
N1
N1 \ufffd 1 r
2
1 þ
N2 \ufffd n2
N2 \ufffd 1
r22
n2
þ N3 \ufffd n3
N3 \ufffd 1
r23
n2n3
r2T þ
r21
N1 \ufffd 1
ð12aÞ
n1 ¼ r
2
1 þ r22 þ r23
r2T
ð12bÞ
These solutions have to be rounded to the nearest upper
integer 1V n1VN1.
Example 4 (Determination of cobalt in nickel cathodes).
Assume that the size of a lot consisting of cathode nickel
is 25 tonne and according to the specifications, the cobalt
content must not exceed 150 g/tonne. Average weight of
the cathode plates produced is 50 kg, and the plates are cut
approximately into 50-g pieces before packing into```