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# Practical applications of sampling theory

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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 [5] and Cochran [7]. 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 [5] 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 [5]. 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