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TECHNICAL REPORT TR 5307 First edition 1991-12-01 Solid fertilizers - Derivation of a sampling for the evaluation of a large delivery plan Ma tibes fertilisan tes solides - Fondements theoriques du plan d%chantNonnage destinb A Mvaluation d’une grosse livraison Reference number ISO/TR 5307: 199 (E) ISO/TR 5307:1991(E) Contents Page Scope References Notation and Symbols Preliminary hypotheses Principle of the sampling plan 5.1 General Theory of the sampling plan 5.2 Information 5.3 What is determined 6.1 Definitions 6.2 Determination by the proposed method of limits 6.3 Use of two non-central 6.4 Determination Practical 7.3 Simplified procedure for the determination 8.2 Simplified of N and N’ 15 15 calculation Examples of calculations 8.1 Calculation 10 of N and N’ 11 7.1 Basic information 7.2 Calculation t distributions 15 when N’ > 30 16 17 by the complete process 17 calculation 20 Effect of the values of the various Parameters on N and K 20 10 Evaluation of a delivery 22 Annexes A Table of values of the standardized normal variable u as a function of P 24 B Table of values of (1 - a2)/a2 25 ISO 1991 All rights reserved No part of this publication may be reproduced or by any means, electronie or mechanical, including photocopying Permission in writing from the publisher International Organlzation for Standardization Case Postale 56 l CH-121 Geneve 20 l Switzerland Printed in Switzerland ii or utilized in any form and microfilm, without ISOITR 5307:1991 (E) Foreword ISO (the International Organization for Standardization) is a worldwide federation of national Standards bodies (ISO member bodies) The work of preparing International Standards is normally carried out through ISO technical committees Esch member body interested in a subject for which a technical committee has been established has the right to be represented on that committee International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization The main task of technical committees is to prepare International Standards, but in exceptional circumstances a technical committee may propose the publication of a Technical Report of one of the following types: - type 1, when the required support cannot be obtained for the publication of an International Standard, despite repeated efforts; - type 2, when the subject is still under technical development or where for any other reason there is the future but not immediate possibility of an agreement on an International Standard; - type 3, when a technical committee has collected data of a different kind from that which is normally published as an International Standard (“state of the art”, for example) Technical Reports of types and are subject to review within three years of publication, to decide whether they tan be transformed into International Standards Technical Reports of type not necessarily have to be reviewed until the data they provide are considered to be no longer valid or useful lSO/TR 5307, which is a Technical Report of type 3, was prepared Technical Committee ISO/TC 134, Fertilizers and seil conditioners by This document is a type Technical Report lt is not envisaged that it will be published as an International Standard lt gives the mathematical derivation of the sampling plan specified in ISO 8634 Annexes A and B are for information only ISO/TR 5307:1991(E) Intrduction Within the framework of its work on sampling, Technical Committee 134 has, through its subcommittee 2, "Fertilizers and Soil Conditioners" carried out statistical studies on various sampling Plans which may be used to assess large deliveries of fertilizers This work complements currently under prebaration, and other Standards for fertilizers, provides the theoretical background necessary to appreciate fully the report (type 3), which is requirements of those Standards This technical different from the international Standards usually produced by ISO/TC 134/SC4, is intended to act as a complement to them, as a basis for the sampling of fertilizer deliveries Esch country has its own regulations applicable to the fertilizer trade; an official department is responsible for carrying out Checks regarding If these regulations are violated, application of the regulations sanctions may be taken against those responsible for placing the fertilizer on the market in that country In the case of an imported it is the representative of the manufacturing Company in the delivery, or the importer who is considered by the relevant authorities to country, be responsible for the declared contents shown on labels or other documentation accompanying the fertilizers ISO 8634 concerns the case of an importer who resells, on his own a large amount of fertilizer received from abroad After responsibility, unloading, this delivery is resold in smaller lots to traders (dealers or fartikr cooperatives) who will themselves be direct suppliers to farmers it is the importer whose name is associated with In the case in question, and it is therefore he who will be considered by the the fertilizer; retailers and users to be responsible for the declared contents ISO 8634 is designed rules for: a) sampling (i.e b) acceptance and both apply iv for acceptance the sampling (the acceptance to the bulk delivery inspection It determines plan); or rejection imported of the delivery); the ISO/TR 5307:1991 (E) The location of the acceptance inspection, as defined in ISO 8634, in the chain of transactions tan be represented by the following diagram: Country Importing of manufacture Seller- Importer - country Retailers -,Farmers Official inspections of lots resold in accordance with national regulations This page intentionaiiy left blank TECHNICAL ISO/TR 5307:1991(E) REPORT - Derivation of a sampling Solid fertilizers evaluation of a large delivery plan for the Scope This Technical the definition Report presents of the sampling the sampling theory which h s resulted plan described in ISO 8634 in If l The sampling plan is applicable to a large delivery of more than 250 t of fertilizer supplied to another Party, for resale, on his responsibility, in small lots, each of which would be subject to own legislation By large (5,000 period amount is understood, for example, a full boat-load t, 10,000 t or more) thus corresponding to a relatively of manufacture, but the theory applies to any delivery long of 250 t or more References ISO 8157: 1984, Fertilizers ISO 8634:.'1, Notation and soil conditioners - Vocabulary Solid fertilizers - Sampling plan for the evaluation of a large delivery and symbols The following Symbols appear in this meanings assigned to them below Technical Report and have the Actual mean value and Standard deviation between sampling units delivery in the Mean value and Standard deviation between sampling units in a delivery acceptable quality l) To be published of just ISO/TR 5307:1991 (E) Mean value and Standard deviation between sampling units in a delivery unacceptable quality of just Mean and Standard deviation, of two lots which tan be respectively, considered by the importer to be of the Same quality u Number of sampling delivery N Number of sampling units to be selected during the sampling of the delivery (Increments) N r units in the Number of analyses to be carried out on the N increments during the inspection of the delivery NR Number of sampling units contained in the smallest lot presented for resale k Number of increments to be combined into each aggregate Sample for analysis 22 Number of sampling units which will be mandatorily selected during the official sampling of a lot of NR sampling units x Mean value found by analysis after the selection of n sampling units from a lot sampling units of S’ with the aid of the of g/ J-k Nr analyses, where CI is the Standard deviation between the sampling units in the delivery xi Analytical result Sample of rank i Estimate obtained on the Estimate delivery of the mean value of the with the aid of the N' analyses Declared nutrient value e.g of a plant in the fertilizer delivery Official inspection limit value which depends on the declared value(l)) It may be equal to D or less than D by a prescribed tolerante which may depend on the size of the lot Sold ISO/TR 5307:1991(E) Probability that the mean value of n sampling units is lower than the official limit value (L), just acceptable by the importer rr Probability that the mean value of n sampling units is lower than the official limit value (L), just unacceptable by the importer a Probability of rejection just acceptable quality risk) of a delivery of (Sellers or producer's Probability of acceptance just unacceptable quality consumer's ff sk) of a delivery of (importe& or L- a normal variable Value of the standardized such that Pr u > ulwr ] equals a L- r normal variable Value of the standardized such that Pr u > ulmr ] equals 2-r a r ‘1-a normal Value of the standardized variable such that Pr[u > ulBa] equals -a Value of the standardized normal variable such that Pr[u > u Jg equals ß Non-centrality Parameter Calculation coefficient on n, the risk levels probability levels r, which is dependent a and ß and the and r,T Constant factor dependent on N' which represents the uncertainty associated with estimate of the Standard deviation tO Value of the non-central Student ratio corresponding to the level of probability non-centrality Parameter equal to JN UI-r a fn BO Limit value from tO of the estimate calculated the for a ISO/TR 5307:1991 (E) A,B Calculation estimation F - Preliminary intermediates used during of the lot after analysis the Calculation intermediate used to facilitate the calculation of k and N hypotheses Thesampling plan serial correlation has been dra wn up on the assumpt ion that there of the del ivery between the success i ve units is no The N units inspected are selected at random from the delivery, each unit having the Same Chance of being selected, and the N groups of k units made up at random from the N It is also understood that the lots made up by the importer represent a random Sample from among the U bags of the delivery and that the increments taken from a lot by the authorities responsible for the inspection are taken at random from the lot In the subsequent theory, it is assumed that a Single plant nutrient of interest or that, if this is not the case, each plant nutrient is considered separately It is also assumed that the fertilizer is packaged although similar arguments will also apply to products in bulk The analytical sampling error error is considered Finally, it is assumed once.and for of data from production and dispatch countries: to be negligible in relation is to the all, as has been shown by the studies inspection carried out in various a) that the mean concentration of a certain component or value in the sampling units (e.g bag) constituting a definite lot of fertilizer shall be considered as a random quantity which obeys a normal distribution; b) that the distribution least for sufficiently Principle of the ~ampling of this random quantity does not large lots, on their size depend, plan 5.1 General The sampling plan described and N', which depend on: in ISO 8634 defines a) the legal requirements of the importing for the value and the size of the smallest inspected); b) the risks which the importer a pair of numbers, country (acceptable lot which tan be N limit accepts for the NOTE - It should be remembered that it is intended inspection of the delivery received by the importer, and not for the lots resold by the Same importer at ISO/TR 5307:1991 (E) tan be applied even for small values comparison with a graphic resolution distributions Allowing this x - Ks has: then approximation, of N' (down to N' = 5) by of the use of non-central Student x and s are independent and p - -aKg a mean of: J-k and a variance r ( a = -3 with k + (1 -az)K2 N'- I' is the Euler function of a tends to the risk of the first (wrongly) an acceptable ul-ra 17 1) NOTE The value Considering rejecting which: t1 N e N N'- L( where of: z as N' tends to infinity type, there is a probability a of delivery, that is, a delivery-for = L P- - E.3 J-n be equal to a value In this limit case 53 - Ks will deviation of-X - Ks) of (X - Ks) - ulVa (Standard Q given by -Q = mean - Thus : aKcr r- Q -=P- r'1-a o + (1 -aZ) - E.4 K2 ~where ulma is the standardized the-probability a Eliminating p between Ul-r J-n a normal these variable two equations aKq - N fk t 12 - + (1 - a*)K* + (1 - a*)K* ' to E.3 and E.4 gives: Q depends on the unknown Standard deviation, Chosen such that -Q = L This value is given K =a corresponding unless by: 1 * the value E.5 of K is ISO/TR 5307:1991 (E) In the same way, for the probability of accepting a delivery for which: yr r P- y-r which, for that is, =L r R = ,L risk of the second tvpe, calling ß the (wrongly) a just unacceptable delivery aKa* ul-ßgr', g = L, requires the (1 - a2) pp following - E.6 Equating E.5 and E.6 gives: (y-r a U1-r r > J Ln (u - + (1 - a2)K2 + '1-ß > [ N* l-a Deducing from E.7 the value of the expression and substituting back into (E.5) gives: K-l Y-a !i- L ,a J n Replacing Cul-r hence r + ul-b U1-r a u1-a + to the power "l/Z" - E.8 ‘1-ß E.7 gives: Cul-a + u1-ß > k-, r n + k (l-a2) Nt a1 [ ul-a - u1-r a2 n * r + '1-ß '1-r (as kN’ = IV): (u l-r N = (Ul a + u l-IJ ) a - - '1-r r >2 that 1-a2 [ - a - U1-r > r numbers, )1 l-a na2 N and N' are to be whole r Cul-r -1 CU n Given fact: raised - E.7 J K by E.8 in the equation - U1-r a U3:r I2 this U1-r r + ul-ß '1-r a becomes in - E.9 n 13 ISOITR 5307:1991(E) by definition Moreover, - a.2 is only a2 5, and consequently of N' This relation has been calculated and is for values of N' between and 30 Consequently, the inequality MV' g (N') And therefore, Chosen, such or k f E.9 tan - a2 in the in annex B form: ’ for each whole value of k, that this inequality is just so the the whole satisfied choice of N' is value of N' tan be restricted to those a a* Having shown in the table be expressed (N’) However, k must be positive, values for which a function - E.10 calculated the limit table in annex B allows of N', i.e N'0 the - a* with of a2 determination of the value the aid of E.lO, smallest possible The smallest values of N' greater inequality E.9 are then determined than N'O which will satisfy the with the Chosen values of k N is of N' and k deduced from the known value The calculation 6.4.2 SimplXied cafculation becomes quite large simplified as soon as N' It is then possible to approximate 1-a2to a* to use the formula: 5-r - '1-r a 1* (1+ K2-1 r where K2 k n [ 14 Y-r a 5-ß - + Y-r r U1-a +u1-ß u1-~ tan be greatly (i.e more than and it N' is the 30) sufficient value ISO/TR 5307:1991 (E) Practical procedure 7.1 Basic for the determination of N and N’ information Local legislation constituting the size N defines n which is Sample required for the minimum number of bags for inspecting the quality of a lot in agreement with the exporter, fixes the values The importer, probability levels 2, 1, and rr; these values, although expressed in percentage form have to be converted into fractions calculation purposes for of the for 7.2 Calculation 7.2.1 Table give the values of u (standardized variable) corresponding to the probabilities ra, and l-ß i.e Ul-y 7.2.2 ’ U1-r r - a2 a2 and u ’ u1-a To determine ( Using a >0 a - '1-r of N', '1-r 10, determine + u1-tl- ul-r r a the value calculate: r [ u1-a table the minimum value U1-r = 1-E of N' such that - a2 corresponding to this value of N' is just a2 calculated above Taking N'0 to be that particular this is the smallest possible value for N' 7.2.3 Leaving - a2 undetermined, normal rr, l-a calculate the less ratio than value of Al'; the.expression: a2 F = n- With the ( '1-a - + u1-ß_) aid for N' = N'g, each of these of the table N U1-r a - U1-r r 1 - a2 ( u1-a- U1-r r a2 +u l-ß U1-r a in annex B, determine the values N'0 + lN'0 + etc and calculate for values of N' the corresponding values hence of k as k is the whole The calculation (N'm) greater part of (I$+l) ) 1-1 of - a2 a2 of F and andthusNasN=kN' is concluded when a value of N' is obtained than the corresponding value of F 15 ISO/TR 5307:1991(E) 7.3 Simplified N is calculation when N' > 30 calculated Ul-ß, n, ulgol, Draw up the as a function of k from the fixed values of ulgra, ulwrr by the formulae in 6.4.2 , N following table noting that Nr = -and therefore that the k value of N multiple has to be Chosen ITable Transfer Table N' above the calculated of k immediately figure I all the results to table 2: F - a2 k N a2 ? N ? N 0+l Y N 0+2 ? N + With the aid of table 16 2, all possible pairs of N and N' may be obtained ISO/TR 5307:1991 (E) For practical in a simplified application, these possible table (see table 3) pairs of N and N’ may be collected Table Nt N Examples 8.1 of calculations l Calculation by the complete process The smallest lots which the importer will resell 8.1.1 General content levels will be lots of one bag (N = 1) for which particular will be guaranteed, for example, 25 % ammonical nitrogen (L = 25 %) Clearly local regulations stipulate, for lots of bag, that a Sample of (2) of which will be compared bag (n = 1) is taken, the content with the limit (L); and if x < L, the lot of bag will be considered as defective and action will be taken against the importer The importer decides to consider as acceptable a delivery for which the probability, r,, of making up, from this delivery, a defective lot of agrees to sel1 bag would be at most % That is to say that the importer one defective lot of bag out of 100 lots of bag-(i.e r, = % or 0,Ol) In addition, he decides to consider the probability of making up, from bag would be equal to or greater 0,lO) Finally, delivery 0,os) as unacceptable a delivery for which this delivery, a defective lot of than 10 % (i.e rr = 10 % or he accepts the value % as the risk of wrongly accepting a % or which should have been considered unacceptable (i.e ß = - the Seller agrees to the value % as the risk For his Part, delivery wrongly rejected which should have been considered acceptable (i.e ct = % or 0,05) of having a The Problem is to determine the number, N, of bags to be selected from out on these the delivery and the number N' of analyses to be carried bags in Order to achieve all these conditions According to the definitions above, we have: N = n =l L = 25 % ammoniacal nitrogen 17 ISOITR 5307:1991 (E) r r a r = %, hence ul r = 2,326 (see table Al) a = 1,282 = 10 %, hence ul r (see table A.l) r a = S.%, hence u1 a = 1,645 (see table A.l) = 1,645 (see table A.l) B = %, hence u 8.1.2 l ß- Minimum value of N' The equation: Y-Z- -a2 (' = a2 ) r1 r J u1-a gives: - U1-r a + u1-B- Y-r r ?-r a 1-a2- (- = 0,0309 > a2 Table B.l 8.1.3 Calculation In view Shows that of the n, u l-y a this implies N' > 17 i.e N'o = 18 of F values given to the Parameters; ) Ul-~ ) ‘l-a3 and ‘1-8 r [( Ul- a r l-a2 - yr r (u1-a- '1-r a2 ) +U r l-f.3 L- a -1 11 gives: -1 F = 10,8241 1,0899 - (35,2261)] a2 8.1.4 Calculation of k and N ‘For N' = 18: - a2 = 0,0299 a2 hence, F = 295,42 F and I-l= N consequently, 17,4 the whole N = kN' = 306 part of which is part of TJ-ich is 17 and thus k = 17 and For N' = 19: - a2 - = 0,0282 a2 hence, F F = 112,14 and - + = 6,90 the whole Nt consequently, N = 114 and thus k = and ISOITR 5307:1991 (E) These calculations Table 1-a' Nt result F in the values k N listed in table a2 18 0,0299 295,4 17 306 19 0,0282 112,l 114 20 0,0267 72,9 80 21 0,0253 54,5 63 22 0,024l 44,9 66 23 0,023O 38,7 46 24 0,022o 34,4 48 25 0,021o 30,9 50 26 0,0202 28,6 52 27 0,0194 26,6 27 The list in table Stops at N' = 27, A series of pairs, N and N', is thus satisfy the stipulated conditions The choice between the various for which F < Nr obtained (see table 5) which possible pairs, N and N', will such as the respective marginal costs of each of these should be analysed Thus, if 27 bags are selected, If, on the other hand, 46 bags are selected, they could individually so that there would only be 23 analyses to perform grouped in pairs, be depend on practical conditions, sampling and analysis this would be beneficial if the tost of an analysis was limit, or more times as high as that of the taking of one more Sample, but more extensive grouping of increments with a view to reducing the number of analyses would not, in financial terms, be worth the trouble At the ISO/TR 5307:1991 (E) 8.2 Simplified With the calculation Same values 2,326 K = for the Parameters as above, x 1,645 + 1,282 x 1,645 k IJ 1,645 + 1,645 and N 1,645 + 645 (l + g) 2,326 - 11282) ' hence N 9,92 + 16,27 k These equations Table N 27 27 22 44 -l3 20 60 19 76 18 108 17 17 289 Effect of the The following procedure: values equation [ 9-r x2 B -+ X Cin [ l+z Parameters on N and K by the simplified k ’ Ul-r r calculation + Ul+ Ul-r I \U a l-a is of the form: (u l-r a or AX2 + BX + C in (ulwa or AX' + B (with various is obtained a - u1-r r is to say it A -+ of the (2 N =n that table: Nt k lead to the following - u1 r ) (with r +‘* 1-B) A and B > 0) in (with A, 23 and C > 0); A, B and C > 0); (ul r" - u1 r ) -a -r - U1-r r\ ISO/TR 5307:1991(E) Similarly, ( 1Ora U Ul -rr+ K = Ul-a- t Table Table summarizes A the effect Ul-B- u1-a- + ul-ß- + ul-ß t -+B x which is of the form - U1-r,) in ‘(u t l-a- + ul-b - ’ OT’ AX + B’ in '(u 1-r - u 1-r a r of a changing the various Parameters: Variation in the Parameters N (sampling Reduction in the between differente the quality levels: Increases rapidly K tost) (stringency inspection) very Increases 'a and rr Reduction in the sum of risks cx and ß Reduction in the sum of defective percentages corresponding to the two quality levels of the (more rapidly if rr decreases than if r increases B Increases very rapidly and increasingly so if the risks are initially greater Increases relatively slowly (if ß decreases more quick17 than a) Increases rapidly Increases fairly rapidly if z- ,decreases more quic I ly than r, very *a + TT Increase in n Increases linearly as a function of 23 Decreases in Proportion to the Square root of n The effect of the various Parameters is shown in numerical terms in table which gives values of N, N' and K for two values of k based on the simplified calculation It should be noted, however, that each set of values for the Parameters leads to a complete series of values for N and N' all of which will fulfil the requirements of the theory (see 7.2.4 and table 6) 21 ISO/TR 5307:1991 (E) Table - Effect of N;N' and K of the various Parameters on the For k = al) n 1 1‘ 1 1 1 5 10 10 10 ') '1 1 1 5 5 1 1 ßl> Ta11 r,l) N 5 10 10 5 10 5 10 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 095 0,5 10 "5 10 10 10 10 10 5 10 56 56 25 25 98 98 45 ' 45 20 20 41 ~' 41 19 19 27 '27 14 14 129 129 63 63 305 305 220 220 91 91 The values 10 Evaluation are expressed values For k = N' _SK N N' K 2,03 1,82 1,93 1,98‘ 1,74 2,ll' l,93 1,80 1,85 0991 0,81 0,67 0,64 0,55 205 90 350 160 65 150 65 90 50 280 -125 525 370 140 41 18 70 32 13 30 13 18 10 56 25 105 74 28 4,54 4,07 4,31 4,42 3,90 4,72 4,31 4,03 4,13 2,03 “1,82 1,49 1,44 1,23 as percentages (see 7.1) of a delivery For each plant nutrient the N' analysis results following intermediate guaranteed a Student obtained, calculating values A and B: test is applied, using successively the X-X)*andB= X-L i=l i LÄ- where X is the mean of the places; analytical Xi is the rank i; result L is analytical the.official limit results obtained as defined calculated on the in clause to decimal aggregate Sample of ISOITR 5307:1991 (E) The delivery does not comply with the evaluation criteria Chosen for given plant nutrient if the value B corresponding to this plant nutrient is less than the value BO which is given, as a function of the number of analyses N', of N increments and of a U1ora and ** XL v6 ? corresponds to the limit corresponding equal to: The value B is non-central to a probability then calculated level Student ratio CJ, with the non-centrality from the equation: Parameter ISO/TR 5307:1991 (E) Annex Table A-1 - Values of the standardized A normal variable u as a function of P ,000 ,001 ,002 ,003 ,004 ,005 ,006 ,007 ,008 900 901 $02 PO3 904 2,3;63 2,0537 1,8808 1,7507 3,0902 2,2904 2,0335 1,8663 1,7392 2,8782 2,2571 2,0141 1,8522 1,7279 2,7478 2,2262 1,9954 1,8384 1,7169 2,6521 2,1973 1,9774 1,825O 1,706O 2,5758 2,1701 1,960O 1,8119 1,6954 2,5121 2,1444 1,9431 1,7991 1,6849 2,4573 2,1201 1,9268 1,7866 1,6747 905 906 YO7 $08 909 1,6449 1,5548 1,4758 1,4051 1,3408 1,6352 1,5464 1,4684 1,3984 1,3346 1,6258 1,5382 1,4611 1,3917 1,3285 1,6164 1,5301 1,4538 1,3852 1,3225 1,6072 1,522O 1,4466 1,3787 1,3165 1,5982 1,5141 1,4395 1,3722 1,3106 1,5893 1,5063 1,4325 1,3658 1,3047 310 Pl1 912 913 Pl4 1,2816 1,2265 1,175o 1,1264 2,0803 1,2759 1,2212 1,170o 1,1217 1,0758 1,2702 1,216O 1,165O 1,117o 1,0714 1,2646 1,2107 1,1601 1,1123 1,0669 1,2591 1,2055 1,1552 1,1077 1,0625 1,2536 1,2004 1,1503 1,1031 1,0581 ,15 316 917 918 919 1,0364 0,9945 0,9542 0,9154 0,8779 1,0322 0,9904 0,9502 0,9116 0,8742 1,0279 0,9863 0,9463 0,9078 0,8705 1,0237 0,9822 0,9424 0,904o 0,8669 1,0194 0,9782 0,9385 0,9002 0,8633 920 921 922 923 924 0,8416 0,8064 0,7722 0,7388 0,7063 8,8381 0,803O 0,7688 0,7356 0,7031 0,8345 0,7995 0,7655 0,7323 0,6999 0,831O 0,7961 0,7621 0,729o 0,6967 925 926 927 928 929 0,6745 0,6433 0,6128 ' 0,5828 0,5534 0,6713 0,6403 0,6098 0,5799 0,5505 0,6682 0,6372 0,6068 0,5769 0,5476 930 931 932 933 934 0,5244 0,4959 ~ 0,4677 0,4399 0,4125 0,5215 0,493o 0,4619 0,4372 0,4097 $35 936 937 938 939 0,3853 0,3585 ~ 0,3319 0,3055 ~ 0,2793 ,40 941 942 943 944 945 ,46 947 948 949 P For ,009 ',OlO 2,4089 2,0969 1,911o 1,774s 1,6646 2,3656 2,0749 1,8957 1,7624 1,6546 2,3263 2,0537 1,8808 1,7507 1,6449 999 38 997 996 995 1,5805 1,4085 1,4255 1,3595 1,2988 1,5.718 1,4909 1,4187 1,3532 1,293o 1,5632, 1,4833 1,4118 1,3469 1,2873 1,5548 1,4758 1,4051 1,3408 1,2816 994 993 32 31 990 1,2481 1,1952 1,1455 1,0985 1,0537 1,2426 1,1901 1,1407 1,0939 1,0494 1,2372 1,185O 1,1359 1,0893 1,045o 1,2319 1,180O 1,1311 1,0848 1,0407 1,2265 1,175o 1,1264 1,0803 1,0364 39 1,0152 0,9741 0,9346 0,8965 0,8596 1,OllO 0,9701 0,9307 0,8927 0,856O 1,0069 0,9661 0,9269 0,889O 0,8524 1,0027 0,9621 0,923o 0,8853 0,8488 0,9986 0,9581 0,9192 0,8816 0,8452 0,9945 0,9542 0,9154 0,8779 0,8416 984 983 0,8274 0,7926 0,7588 0,7257 0,6935 0,8239 0,7892 0,7554 0,7225 0,6903 0,8204 0,7858 0,7521 0,7192 0,6871 0,8169 0,7824 0,7488 0,716O 0,684O 0,8134 0,779o 0,7454 0,7128 0,6808 0,8099 0,7756 0,7421 0,7095 0,6776 0,8064 0,7722 0,7388 0,7063 0,6745 979 0,665l 0,6341 0,6038 0,574o 0,5446 0,662O 0,6311 0,6008 0,571o 0,5417 0,6588 0,628O 0,5978 0,5681 0,5388 0,6557 0,625O 0,5948 0,5651 0,5359 0,6526 0,6219 0,5918 0,5622 0,533o 0,6495 0,6189 0,5888 0,5592 0,5302 0,6464 0,6158 0,5858 0,5563 0,5273 0,6433 0,6128 0,5828 0,5534 0,5244 0,5187 0,4902 0,4621 0,4344 0,407o 0,5158 0,4874 0,4593 0,4316 0,4043 0,5129 0,4845 0,4565 0,4289 0,4016 0,5101 0,4817 0,4538 0,4261 0,3989 0,5072 0,4789 0,451o 0,4234 0,3961 0,5044 0,4761 0,4482 0,4207 0,39.34 0,5015 0,4733 0,4454 0,4179 0,3907 0,4987 0,4705 0,4427 0,4152 0,388O 0,4959 0,4677 0,4399 0,4125 0,3853 ~ 0,3826 0,3558 ' 0,3292 ~ 0,3029 ~ 0,2767 0,3799 0,3531 0,3266 0,3002 0,274l 0,3772 0,3505 0,3239 0,2976 0,2715 0,3745 0,3478 i 093213 ~ 0,295o I 0,2689 0,3719 0,3451 0,3186 0,2924 0,2663 0,3692 0,3425 0,316O 0,2898 0,2637 0,3665 0,3398 0,3134 0,2871 0,2611 0,3638 0,3372 0,3107 0,2845 0,2585 0,3611 0,3345 0,3081 0,2819 0,2559 0,3585 0,3319 0,3055 0,2793 0,2533 964 363 962 961 960 0,2533 0,2275 0,2019 0,1764 0,151o 0,2508 0,225O 0,1993 0,1738 0,1484 0,2482 0,2224 0,1968 0,1713 0,1459 0,2456 0,2198 0,1942 0,1687 0,1434 ~ 0,243O 0,2173 0,1917 0,1662 0,1408 0,2404 0,2147 0,1891 0,1637 0,1383 , 0,2378 0,2121 0,1866 0,1611 0,1358 0,2353 0,2096 0,184O 0,1586 0,1332 0,2327 0,207O 0,1815 0;1560 0,1307 0,2301 '0,2045 0,1789 0,1535 0,1282 0,2275 0,2019 0,1764 0,151o 0,1257 959 ,58 957 956 955 OJ257 0,1004 0,0753 0,0502 0,0251 0,1231 0,0979 0,0728 0,0476 0,0226 0,1206 0,0954 0,0702 0,,0451 0,0201 0,1181 0,0929 0,0677 0,0126 0,0176 0,1156 0,0904 0,0652 0,0401 0,015o 0,113o 0,0878 0,0627 0,0376 0,0125 0,1105 0,0853 0,0602 0,0351 0,0100 0,108O 0,0828 0,0577 0,0326 0,0075 0,1055 0,0803 0,0552 0,0301 0,005o 0,103o 0,0778 0,0527 0,0276 0,0025 0,1004 0,0753 0,0502 0,0251 o,oooo 954 953 example: for u = 1,2536 24 a probability of 10,5 % (P = 0,105) 988 37 986 35 32 981 980 978 977 975 ,74 973 972 971 970 969 968 967 966 365 32 951 950 ISO/TR 530711991 (E) Annex B B-1 - Values Table of (1 - -a*)/a* - N a (1 - a*>/a* 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ,940o ,9515 ,9594 ,965O ,9693 39727 ,9753 ,9776 ,9794 ,981O ,9823 ,983s ,9845 ,9854 ,9862 ,9869 ,9876 ) 9882 ,9887 ,9892 ,9896 ,990l ,9904 ,9908 ,991l ,9914 ,1317 ,1045 ,0865 ,0738 ,0643 ,057o ,0512 ,0464 ,042s ,0392 ,0363 ,0338 ,0317 ,0299 ,0282 ,0267 ,0253 ,024l ,023O ,022o ,021o ,0202 ,0194 ,0187 ,018O ,0174 25 ISO/TR 5307:1991 (E) UDC 631.8519.243 Descriptors: fertilizers, Price based on 18 pages sampling, sampling tables, statistical distribution, Standard deviation, rules of calculation