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# Practical applications of sampling theory

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barrels for shipment. For the analysis, these 50-g pieces are taken as primary samples, and from them a 1-g sample is dissolved for the cobalt determination. The cost of one analytical determination is 12 o, that of taking one primary sample (50-g piece from a given plate) is 2 o. The standard deviations have been estimated as between- plates standard deviation s1 = 35 g/tonne, within-plate standard deviation (standard deviation of 50-g pieces taken from a single plate) s2 = 15 g/tonne, and the standard deviation between 1-g samples taken from a single 50-g piece, s3 = 3.3 g/tonne. Optimize the analytical procedure so that the standard deviation of the lot mean does not exceed the value of 5 g/tonne.. Sampling is carried out at three different error-generating levels. Following values apply to these: level 1: N1 = 25000 kg/50 kg = 500, s1 = 35 g/tonne, c1 = 0 o, n1=? level 2: N2 = 50 kg/50 g = 1000, s2 = 15 g/tonne, c2 = 2 o, n2=? level 3: N3 = 50 g/1 g = 50, s3 = 3.3 g/tonne, c3 = 12o, n3=? At level 1, the unit cost is practically zero. At this level, the only sampling procedure is the decision made from which plates the 50-g primary samples should be taken. Solution. Eq. 10 gives the results: n3 = 0.09, by apply- ing the constraints we have to select n3 = 1; n2 = 0, we have to select n2 = 1. For the level 1, the following results are obtained: n1= 53.3 (Eq. (12a)) or n1= 58.4 (Eq. (12b)). The approximate solution slightly overestimates the required number of plates to be sampled, but as long as the size of the sample at any level is, say 10% or less from the lot size, the difference is small. In this case, the following sampling protocol could be used. A 50-g piece is taken from every ninth cathode plate at the packing stage. This gives a total of 55 samples per a lot of 25 tonne. From each 50-g piece, one cobalt x¯ i ni Total cost of the investigation in general case is ct ¼ Xk i¼1 nici ð16aÞ Usually in practice, the costs of sampling and analysis are independent from the strata from which the samples are taken and this equation simplifies to ct ¼ ntc*; if c1 ¼ c2 ¼; . . . ;¼ ck ¼ c* ð16bÞ Optimization of investigation involves the optimal allo- cation of the total number of samples that can be analyzed P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u201394 91 determination is made. The total cost of this inspection scheme is 55(0 o+ 2 o + 12 o) = 770 o. The ex- pected standard deviation of the lot mean (from Eq. (8a)) is 4.9 g/tonne. Fig. 5 shows the Operation Characteristics curve of this inspection scheme. It shows both the producer\u2019s and buyer\u2019s risk by using this inspection scheme. 4.2. Optimization of sampling plan, when the lot consists of strata of different sizes and heterogeneities Sometimes the lot to be investigated consists of well- defined strata of different sizes and heterogeneities; for example, when the batch to be processed is prepared by mixing raw materials of different qualities to achieve the required average composition. This kind of a lot is shown in Fig. 6. Fig. 5. Operation Characteristics of the optimized inspection protocol for cobalt determination. In this figure x-axis shows the true mean value of the lot, and y-axis, the probability that the mean value of the inspection exceeds the specification 150 g/tonne (producer\u2019s risk) or the probability that the inspection value is below the specification (buyer\u2019s risk). The lot consists of k different strata. Quantities needed to design a cost optimal sampling plan are Wi ¼ MLiP MLi ¼ Relative size of the stratum i ð14Þ where MLi, sizes of strata (e.g., as mass or volume; i = 1,2, . . ., k); Ni, relative size of stratum i expressed as the number of potential samples that could be taken from the strata =Ni=MLi/MSi; MSi is the size of samples taken from stratum i; ri, standard deviation of one sample taken from stratum i; ci, cost of one sample analyzed from stratum i; ct, total cost of the estimation of the grand mean of the lot; ni, number of samples taken and analyzed from stratum i; nt, total number of samples analyzed =Sni; xij, analytical results on samples from stratum i (i = 1,2,. . .,k; j = 1,2, . . ., ni); x¯i ¼ Pni j xij ni =Mean of stratum i; x¯¯ ¼Pki¼1Wix¯i= grand mean of the lot. Variance of the lot mean r2x¯¯ ¼ X W 2i Ni \ufffd ni Ni \ufffd 1 r2i ni ð15aÞ If the samples taken are small in comparison to the stratum size (as is usually the case), this equation simplifies to r2¯c X W 2 r2i ; if in all strata nibNi and NiH1 ð15bÞ Fig. 6. Lot consisting of k strata of different sizes, and the quantities needed to optimize the sampling plan. between the strata in an optimal way. Mathematical opti- Example 5 (Optimal design for estimating sulfur balance of a pulp mill). Estimation of sulfur balance for a pulp mill is a difficult task Fig. 7. Sulfur enters the mill in raw materials (water, wood) and in chemicals. The outflowing streams consist of products, wastewater, solid wastes, and atmo- spheric emissions. Initial calculations showed that the mean values of all other streams, except the emissions into atmosphere, could be estimated reliably. Atmospheric emissions comprised about one quarter of the total sulfur. Estimation of atmospheric sulfur emissions in an old pulp mill, like the one where this study was carried out, is difficult and therefore optimization of the emission measurement plan is a challenging task. This is due to the fact that there is a large number of gaseous outlets into atmosphere. These have different mass-flows, concentra- tions of the sulfur compounds are highly variable and sulfur is found in dust and in many gaseous compounds [SO2, P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u20139492 mum can again be derived by assuming that ni is continu- ous. The results given below have been derived assuming that investigation costs are independent from strata (Eq. (16b) valid). Optimization strategy depends on how much information is available. If only the sizes of strata are known and the total cost ct of the investigation is fixed, then the best strategy is to allocate the samples proportionally to the sizes of strata: ni ¼ Wint ð17Þ and nt ¼ ct c* ð18Þ Both nt and ni have to be rounded to integers so that the total cost will not be exceeded. If the unit costs and standard deviations are available, then even better plans can be designed. Laboratories usually follow their costs and, consequently, good cost estimates are available. The standard deviations can be estimated either by using Gy\u2019s sampling theory, if the material properties needed are available, or experimentally from a pilot study. If the quality control of the analytical laboratory is well planned, it also provides data that can be used for optimi- zation of sampling and analytical procedures. Cost optimal plan for the investigation can be derived in two ways. Either the total cost of the investigation is fixed and the variance of the grand mean of the lot is minimized or the target value for the variance is given and the total cost is minimized. By assuming that Eqs. (15b) and (16b) are valid, that is, in all strata the samples are small in compar- ison to the sizes of strata, and the cost of investigation is independent of strata, the following results can be derived. 4.2.1. Maximum value, cmax, given to the total cost, variance of the lot mean minimized ni ¼ WiriXk i¼1 Wiri cmax c* ð19Þ Here, ni has to be rounded to integers so that the target cost is not exceeded. 4.2.2. Target value, rt, given to the standard deviation of the lot mean, total cost minimized ni ¼ Wirir2T Xk i¼1 Wiri ð20Þ Again, ni has to be rounded to integers so that the required standard deviation of the lot mean will not be exceeded, i.e., r2x¯¯Vr 2 T. H2S, CH3S, (CH3)2S, and (CH3)2S2]. In optimization, all different emission sources and sulfur-containing compounds analyzed