On One-Sided Tolerance Intervals of Normal Distribution With Unknown Parameters 1

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1 Joural of the Alied Mathematics, Statistics ad Iformatics (JAMSI), 1 (005), No. 1 O Oe-Sided Tolerace Itervals of Normal Distributio With Ukow Parameters 1 IVAN JANIGA AND IVAN GARAJ Abstract I the aer there is derived a exact formula for comutig oe-sided tolerace factors of a ormal distributio with both mea µ ad variace σ ukow. There are also icluded four aroximatios by meas of which the oe-sided tolerace factors ca be comuted roughly. Some of the ossible alicatios of oe-sided tolerace factors i the samlig isectio are reseted o examles. Mathematics Subject Classificatio 000: 6H10 Additioal Key Words ad Phrases: tolerace iterval, tolerace factor, exact comutatio, aroximate comutatio 1. INTRODUCTION Let radom samle X 1,X, K,X be take from distributio N ( µ, σ ). Parameters µ ad σ are ukow. Their ubiased estimators are X ad S, where X = 1 i = 1 X comuted by formulas i 1 x = 1, S = ( 1) = ( X i i X ). Estimates of the arameters are 1 x i i= 1 ( x i is a value of the variable a s 1 = 1 i= 1 ( x i x) X i, thus the ith measured value). We will fid itervals, which with cofidece fractio ( 0 < < 1 ) of values of the distributio (1) 1 ( 0 < < 1) cover at least the N( µ, σ ). Such itervals are called 100 % tolerace itervals. The exlaatio i more detail ca be foud i [1], [] ad 1 This cotributio was suorted by the Grat VEGA roject No. 1/147/04 Advaced statistical techiques ad decisio makig i the quality imrovemet rocess from the Sciece Grat Agecy of the Slovak Reublic. 87

2 I. Jaiga, I. Garaj [3]. Tolerace itervals ca be two-sided or oe-sided. Widesread tables of two-sided tolerace factors ca be foud i moograhs [4] ad [5]. I the followig text we are oly dealig with oe-sided tolerace itervals.. ONE-SIDED TOLERANCE INTERVALS Right-had tolerace iterval is the iterval (, X + ks ) P[ P X < X + ks) ] = 1, for which is valid ( () ( X is a radom variable ormally distributed with mea µ ad variability σ, its realizatio x is a value of ay ideedet observatio take from the oulatio, from which the radom samle X,X tolerace iterval () is (, ks), K,X 1 was take.). A realizatio of the right-had x +. Left-had tolerace iterval is the iterval ( X k S, ), for which is valid P [ P( X X ks ) ] = > 1 (3) ( s, Its realizatio is x k ). The value of the factor k is determied so that the tolerace itervals with the cofidece distributio ( µ, σ ) N. Number oe-sided tolerace factor. 1 cover at least fractio of the values of the 1 amed the cofidece level ad costat k is the 3. EXACT COMPUTATION OF TOLERANCE FACTORS Let us cosider radom variables X -µ S Y + δ Y =, V =, t = σ σ V The radom variable Y has a stadard ormal distributio N (0,1). It is ideedet from the radom variable V, which has = χ -distributio with 1 degrees of freedom. The radom variable t has a o-cetral t distributio with a arameter of ( < < o-cetrality δ δ ). Its robability desity fuctio [6] is 88

3 f O Oe-Sided Tolerace Itervals of Normal Distributio With Ukow Parameters ( t, δ ) = where Γ 1 e π δ ( + t ) t y y e 0 tδ + t 1 x Γ ( ) = x e dx is the gamma fuctio ( > 0). 0 dy < t < The o-cetral t distributio with the desity fuctio (5) will be deoted (, δ ) Whe δ = 0 the distributio t (, δ ) equals the cetral distributio () The mea E [ t (, δ )] ad variace [ t (, δ )] by [7] E D -1 Γ [ t (, δ )] = δ = bδ 1 Γ t. D of the distributio (, δ ) (5) t. t are give ( > 1) (6) ( > ) (7) ( 1) [ t (, δ )] = (δ + 1) b δ where b Let us deote the lower + 1 Γ = ( = 1) (8) Γ 100 ercetage oit (0 < 1) < of the distributio t (, δ ) by sig t (, δ ). It follows from (5) that for ay real umbers t ad δ holds f ( t, δ) f ( t, δ) = (9) t (, δ ) = t1 (, δ ) (10) 89

4 The most widesread tables of ercetiles of o-cetral t-distributios ca be foud i the moograh [8], where they are comuted with accuracy to five decimal laces for = 0,01; 0,05; 0,05; 0,10; 0,0; 0,30; 0,70; 0,80; 0,90; 0,95; 0,975; 0,99, = 1 (1) 60 ad δ = 0,1 (0,1) 8,0. I. Jaiga, I. Garaj For the right-sided tolerace iterval (, X + ks ) is X + ks -µ Z = P ( X < X + ks ) = Φ (11) σ where Φ is the distributio fuctio of a stadard ormal distributio N (0,1). For give 0 1) where ( < < the iequality Z is equivalet to X u is the lower t + 1 = e ks µ u σ π dt 100 ercetage oit of N (0,1) determied so that for the give cofidece coefficiet P Z ) = 1 (1). The tolerace factor k is 1 it is fulfiled the coditio (. By usig cosecutive modificatios ad utilize (4) we obtai P The radom value (X µ) σ u ( ) X + ks µ + σu = P k = S σ ( t k ) = = P 1 (13) X µ σ u t = has a distributio t ( 1, u ). S From relatios (10) ad (13) it follows that t k = ( 1, u ) t (,u ) 1 1 = (14) 90

5 For the left-sided tolerace iterval ( ks, ) P O Oe-Sided Tolerace Itervals of Normal Distributio With Ukow Parameters X we will also get ( t k ) = 1, where t has a distributio t ( 1,u ) ad k is agai give by (14). The exact comutatio of the oe-sided tolerace factors k is closely related with the exact comutatio of ercetiles of o-cetral t-distributio. However tables of ercetiles of o-cetral t-distributio [8] caot be used to solve this roblem because they do ot cotai a arameter of o-cetrality δ = u. Therefore the comutatio of the tolerace factors k was realised i the rogram system Mathematica [9], which cotais the ercetile fuctio (, δ ) t. But for large ( 140) Mathematica is ot able to comute the values of ercetiles by this way. This roblem was solved by usig stadard methods of umerical itegratio of the desity (5). The comutatio is extremely time cosumig whe the arameter of o-cetrality δ = u 80. The comuted values were rouded u to four decimal laces. 4. APPROXIMATE COMPUTATION OF ONE-SIDED TOLERANCE FACTORS There exist a great umber of aroximatios of the oe-sided tolerace factors [1], [10] ad [11]. The Wallis s aroximatio [1] is the best kow ad u till ow the most used. It was derived o the basis of the aroximatio of the statistic X + ks by the ormal distributio give by the relatioshi σ N µ + kσ, σ k +. The oe-sided tolerace factor k is where A u1 1 ( 1) u k, B = u + u1 u A AB = ad the ercetiles u a (15) u 1 ca be comuted by the relatioshi (1). Slightly better results are give by Jeett s ad Welch s aroximatio [13] by meas of the ercetiles of the o-cetral t-distributio 91

6 I. Jaiga, I. Garaj where δb t (, δ ) + u b + ( 1 b )( δ u ) ( 1 b ) b u (16) b is give by (8). For a small δ it is coveiet the aroximatio by va Eede [11], which was derived by meas of Corish-Fisher exasio [14] 1) + t (,δ ) u 3 u + u + 4 5u u u + ( = 4 3 u + 1 u 4u + 1u + 1 u + 4u u 1 u 1+ + δ + + δ δ δ δ 3 (17) Neither of the above metioed aroximatios ca be cosidered to give good results i geeral. This o-availability does ot cotai Akahira s aroximatio [15]. The ercetiles of the o-cetral t-distributio = t (, δ ) ca be foud by solvig the followig equatio where b x δ 1+ x ( 1 b ) = u 4 b is give i (8) ad x 3 x ( u 1) [ 1+ x ( 1 b )] ( = 1) (18) u i (1). The aroximatio (18) yields good results eve if is small. Whe 00 there are may cases where the error is less tha There is a disadvatage that the equatio (18) ca be solved for ukow x oly by usig umerical methods. I moograh [0] to be ublished, tables 1a, 1b, 1c till 16a, 16b ad 16c cotai the values of oe-sided tolerace factors k rouded u to four decimal laces for cofidece levels 1 = 0,001; 0,005; 0,01; 0,05; 0,05; 0,10; 0,5; 0,50; 0,75; 0,90; 0,95; 0,975; 0,99; 0,995; 0,999; 0,9999, for the values of = 0,55; 0,55(0,05) 0,70; 0,75; 0,75(0,05) 0,90; 0,91(0,01) 0,97; 0,975; 0,98; 0,99; 0,991(0,001) 0,999; 0,9999 ad for the samles of sizes = (1) 00; 05(5) 300; 310(10) 400; 45(5) 1000; 1500; 9

7 O Oe-Sided Tolerace Itervals of Normal Distributio With Ukow Parameters 000(1000) 5000; I the last lie ( ) there are the values of quatiles u of the stadard ormal distributio. A abridged versio of the tables 5a, 5b, 5c ad 10a, 10b, 10c are tables umbered by 1 ad resectively. The tables of tolerace factors k ublished u to ow are either so widesread or accurate as those give i the moograh [0]. I [16] they are comuted for four decimal laces ad for cofidece levels 1 = 0,90; 0,95; 0,99; 0,995; 0,999; for the values of = 0,75; 0,90; 0,95; 0,99; 0,995; 0,999 ad for the samles of sizes = (1) 100; 10() 150; 155(5) 00; 0(0) 300, though i may cases the error is already o the third decimal lace. I the most detailed tables [17] the factors are comuted for three decimal laces ad for cofidece levels 1 = 0,005; 0,01; 0,05; 0,05; 0,10; 0,5; 0,50; 0,75; 0,90; 0,95; 0,975; 0,99; 0,995; for the values of = 0,75; 0,90; 0,95; 0,975; 0,99; 0,999; 0,9999 ad for the samles of sizes = (1) 100; 10() 180; 185(5) 300; 310(10) 400; 45(5) 650; 700(50) 1000; 1500; 000; 3000; 5000; ad. Examle 1 (variables accetace samlig) I variables accetace samlig we are give geerally a lower secificatio limit LSL or a uer secificatio limit USL. We assume that measuremets of a quatitative quality characteristic ( x,x, L ) N ( µ, σ ),x 1 are realizatios of a ormal distributio, where ukow arameters µ ad σ ca be estimated by (1). I the case of a uer secificatio limit USL, a lot of roducts are acceted if the samlig isectio gives the result x k s USL +. I the case of a lower secificatio limit LSL, a lot of roducts is acceted if the samlig isectio gives the result x k s LSL of the cofidece coefficiet. I both cases the oe-sided tolerace factor k is tabulated. The choice 1 deeds uo whether the samlig la is beefit for the roducer or for the cosumer. For examle, the cosumer chooses 1 = 0, 05 ad = 0, 90. For = 10, the value of = 0, 7116 k ca be foud i table 1. I the case of the uer secificatio limit USL the lot of roducts is acceted if x + 0,7116s USL ad rejected if x +,7116s > USL 0. If this lot cotais the fractio % 90 % of measuremets uder the USL, the robability of 100 = 1 = I the case of a lower secificatio limit LSL the lot accetace is (1 ) 0,95. 93

8 of roducts is acceted if I. Jaiga, I. Garaj x 0,7116s LSL ad rejected if x 0,7116s < LSL. If this lot cotais the fractio % 90 % of measuremets over the LSL, the 100 = robability of accetace is 1 ( 1 ) = 0,95. Examle (examle 1 cotiued) The quality characteristic of olyester 104 is viscosity which is measured i mpa s (mili Pascal sekuda) ad the temerature is 5 C [18]. The stadard demad for the uer viscosity limit is USL = mpa s. Polyester 104 is delivered i a costat umber of tus whose cotets rereset oe roductio batch. The test results are ot ractically iflueced by errors. A radom samle yielded the = 10 ideedet measuremets: By the W-criterio [19] was affirmed a good coicidece with the ormal distributio (P value is 0,7385). We choose 1 = 0, 05 ad = 0, 90 examle 1. The for = 10, the value of = 0, 7116 just like that give i k ca be foud i table 1. From the measuremets by usig (1) we ca comute x = 943, 8 ad = 3,0111. x + ks = 943,8 + 0,7116 3,0111 = 945,94 < s Whereas there is ot ay reaso to reject the lot of olyester 104. Whe this lot cotais the 100 = of measuremets uder the limit USL =1 000 mpa s, fractio % 90 % 1 = = the it will be acceted with robability (1 ) 1 0,05 0,95. Examle 3 (examle cotiued) For data from examle it is required to comute the right-haded tolerace iterval (, x + k s) with cofidece 1 = 0, 90 which covers the fractio = 0, 99 values from distributio ( µ, ) N σ. The for = 10, the value of = 3, 5317 if k ca be foud i table. The the uer limit is x + ks = 943,8 + 3,5317 3,0111 = 954, 43 ad the whole iterval ( ;954,43) covers at least the fractio 100 % = 99 % of the values of radom samle, take from the above metioed roductio lot, with robability 0,90. O the basis of the 10 values from examle it ca be exected with 94

9 O Oe-Sided Tolerace Itervals of Normal Distributio With Ukow Parameters robability 0, 90, that at least the fractio of the 99 % measuremets take from the lot of olyester 104 will have a viscosity less the 954,43 mpa s. 5. CONCLUSION Tolerace itervals are cosiderable i the statistical quality cotrol. The methods of the statistical quality cotrol are give i [1], []. REFERENCES [1]. JÍLEK, M. Statistické toleračí meze. Praha: SNTL, s. (i Czech). []. HÁTLE, J., LIKEŠ, J. Základy očtu ravděodobosti a matematické statistiky. Praha: SNTL/ALFA, s. (i Czech). [3]. JANIGA, I., MIKLÓŠ, R. Statistical tolerace itervals for a ormal distributio. I Measuremet Sciece Review. ISSN 13, 001, vol. 1, o. 1, [4]. GARAJ, I., JANIGA, I. Dvojstraé toleračé medze re ezámu stredú hodotu a roztyl ormáleho rozdeleia. Bratislava: Vydavateľstvo STU, s. ISBN [Two sided tolerace limits of ormal distributio with ukow mea ad variability]. [5]. GARAJ, I., JANIGA, I. Dvojstraé toleračé medze ormálych rozdeleí s ezámymi stredými hodotami a s ezámym soločým roztylom. Two sided tolerace limits of ormal distributios with ukow meas ad ukow commo variability. Bratislava: Vydavateľstvo STU, s. ISBN [6]. JOHNSON, N. L., WELCH, B. L. Alicatios of the o-cetral t-distributio. I Biometrika, 1940, vol. 31, [7]. HOGBEN, D., PINKHAM, R. S., WILK, B. B. The momets of the o-cetral t-distributios. I Biometrika, 1961, vol. 48, [8]. BAGUI, S. C. CRC Hadbook of Percetiles of No-Cetral t-distributios. CRC Press, Boca Rato, Florida, ISBN [9]. WOLFRAM, S. The Mathematica Book. 3rd ed. Wolfram Media/Cambridge Uiversity Press, ISBN [10]. SAHAI, H., OJEDA, M. M. A comariso of aroximatios to ercetiles of the ocetal t- distributio. I Revista Ivestigacio Oeracioal, 000, vol. 1, o., [11]. EEDEN, va C. Some aroximatios to the ercetage oits of the o-cetral t-distributio. I Revue de l Istitut Iteratioal de Statistique 9, 1961, [1]. WALLIS, W. A. Use of Variables i Accetace Isectio for Percet Defective. Selected Techiques of Statistical Aalysis (EISENHART, C., HASTAY, M. W., WALLIS, W. A. eds.). New York, 1947, McGraw Hill Bokk Comay, [13]. JENNETT, W. T., WELCH, B. L. The cotrol of roortio defective as judged by a sigle quality characteristic varyig o a cotiuous scale. I Joural of the Royal Statistical Society, B 6, 1939, [14]. CORNISH, E. A., FISHER, R. A. Momets ad cumulats i the secificatio of distributios. I Revue de l Istitut Iteratioal de Statistique, 5, 1937, [15]. AKAHIRA, M. A higher order aroximatio to a ercetage oit of the o-cetral t-distributio. I Commuicatios i Statistics, Part B: Simulcitio ad Comutatio, 1995, 4(3), [16]. LIKEŠ, J., LAGA, J. Základí statistické tabulky. Praha: SNTL, s. (i Czech). [17]. ODEH, R.E., OWEN, D.B. Tables for Normal Tolerace Limits, Samlig Plas, ad Screeig. New York, Marcel Dekker, ISBN [18]. GARAJ, I. Preberací lá meraím ri daých jedostraých toleračých medziach. I Statistika, 10, 199, s (i Slovak) [19]. GARAJ, I. W-kritérium a jeho sracovaie a očítači. I 9. medziárodý semiár Výočtová štatistika. ISBN , SŠDS 000, s (i Slovak). [0]. GARAJ, I., JANIGA I. Jedostraé toleračé medze ormáleho rozdeleia s ezámou stredou hodotou a roztylom. Oe Sided Tolerace Limits of Normal Distributio with Mea ad Variability Ukow. (to be ublished). 95

10 I. Jaiga, I. Garaj 96

11 O Oe-Sided Tolerace Itervals of Normal Distributio With Ukow Parameters ,9346 0,1381 0,4748 0,7110 0,9539 1,1079 1,4093 1,7609-0,3493 0,3345 0,639 0,8748 1,197 1,958 1,665,0175-0,1679 0,4439 0,7434 0,986 1,46 1,4198 1,7681,187-0,0659 0,5188 0,8178 1,0607 1,3310 1,5100 1,8707,3018 0,003 0,5749 0,8748 1,109 1,3965 1,5797 1,9500,3937 0,0545 0,6191 0,904 1,1693 1,4493 1,6359,0139,4676 0,0949 0,6553 0,9581 1,095 1,4931 1,686,0669,591 0,177 0,6856 0,9899 1,435 1,5304 1,73,111,5813 0,155 0,7116 1,0173 1,79 1,566 1,7566,1511,664 0,1787 0,734 1,0413 1,986 1,5908 1,7867,1853,6661 0,1991 0,754 1,065 1,314 1,6158 1,8134,157,7013 0,171 0,7719 1,0815 1,3418 1,6383 1,8373,49,738 0,330 0,7879 1,0985 1,360 1,6585 1,8589,675,7613 0,474 0,803 1,1140 1,3770 1,6769 1,8786,899,787 0,603 0,8155 1,18 1,393 1,6938 1,8966,3104,8110 0,71 0,875 1,141 1,4063 1,7093 1,9131,393,839 0,89 0,8387 1,153 1,4193 1,736 1,984,3468,8531 0,98 0,8490 1,1643 1,4314 1,7369 1,946,3630,8719 0,300 0,8585 1,1746 1,446 1,7493 1,9559,3781,8894 0,3394 0,8979 1,174 1,4891 1,8006,0108,4408,96 0,3671 0,976 1,499 1,544 1,8397,057,4886 3,0176 0,3887 0,9511 1,756 1,555 1,8708,0859,567 3,0618 0,406 0,9703 1,966 1,5755 1,8963,1133,5580 3,098 0,408 0,9864 1,3143 1,5948 1,9178,1363,5843 3,188 0,433 1,0001 1,394 1,6114 1,936,1561,6070 3,1551 0,453 1,05 1,354 1,6385 1,9664,1884,6440 3,1983 0,4689 1,0401 1,3737 1,6599 1,990,140,6734 3,34 0,4816 1,0545 1,3896 1,6774,0097,350,6975 3,604 0,49 1,0665 1,4030 1,691,061,55,7176 3,839 0,5011 1,0767 1,4144 1,7047,0401,676,7349 3,3040 0,5089 1,0856 1,443 1,7155,05,806,7499 3,315 0,5157 1,0934 1,439 1,751,069,91,7631 3,3369 0,517 1,1003 1,4406 1,7336,074,303,7748 3,3505 0,570 1,1065 1,4475 1,741,0809,3114,7853 3, ,5318 1,110 1,4538 1,7481,0886,3197,7948 3,3738 0,536 1,1171 1,4594 1,7543,0956,37,8034 3,3839 0,540 1,117 1,4646 1,7600,100,3341,8113 3,3931 0,5439 1,160 1,4694 1,7653,1078,3404,8186 3,

12 I. Jaiga, I. Garaj ,547 1,199 1,4738 1,7701,113,346,853 3,4094 0,5503 1,1335 1,4778 1,7746,1183,3516,8315 3,4167 0,5631 1,1484 1,4945 1,7930,1389,3738,8571 3,4465 0,575 1,1595 1,5069 1,8068,1543,3904,8761 3,4687 0,5799 1,168 1,5167 1,8176,1664,4034,8911 3,486 0,5858 1,175 1,546 1,863,176,4139,9033 3,5005 0,5908 1,1810 1,531 1,8336,1844,48,9135 3,513 0,5950 1,1860 1,5368 1,8398,1913,430,91 3,54 0,6018 1,1941 1,5458 1,8499,06,444,9361 3,5388 0,6071 1,004 1,559 1,8578,114,4519,9471 3,5516 0,6114 1,055 1,5587 1,8641,186,4596,9560 3,560 0,6149 1,097 1,5635 1,8694,45,4660,9634 3,5706 0,6179 1,133 1,5675 1,8739,96,4715,9697 3,5780 0,68 1,56 1,5814 1,8893,469,4901,991 3,6031 0,6343 1,39 1,5897 1,8986,573,5013 3,0041 3,618 0,6416 1,417 1,5997 1,9097,697,5147 3,0196 3,6364 0,6460 1,470 1,6056 1,9163,77,58 3,089 3,647 0,6490 1,506 1,6097 1,908,83,583 3,0353 3,6547 0,6564 1,596 1,6199 1,93,951,541 3,051 3,6733 0,6745 1,816 1,6449 1,9600,364,5759 3,0903 3,7191 TABLE 1 - = 0,90 0,75 0,90 0,95 0,975 0,99 0,995 0,999 0,

13 O Oe-Sided Tolerace Itervals of Normal Distributio With Ukow Parameters ,845 10,58 13, , ,5001 0,486 4,5816 9,587,609 4,58 5,3115 6,438 7,3405 8,094 9,651 11,5663 1,973 3,1879 3,9566 4,6370 5,4383 5,988 7,194 8,5330 1,6978,744 3,3999 3,9814 4,6660 5,1360 6,1113 7,3114 1,5399,4937 3,0919 3,605 4,46 4,6695 5,5556 6,6457 1,4353,337,8938 3,389 3,971 4,3719 5,018 6,6 1,3599,186,7543 3,69 3,786 4,1638 4,9547 5,975 1,304,139,6500 3,1057 3,6415 4,0089 4,7711 5,7084 1,568,0657,5684 3,0113 3,5317 3,8885 4,686 5,5385 1,195,0113,507,9353 3,4435 3,7918 4,514 5,403 1,1883 1,966,4483,875 3,3707 3,711 4,401 5,903 1,1617 1,981,405,8197 3,3095 3,645 4,3410 5,1963 1,1387 1,8954,363,7745 3,57 3,5879 4,735 5,1161 1,1186 1,8669,390,735 3,119 3,5384 4,151 5,0466 1,1008 1,8418,990,7008 3,171 3,4949 4,1639 4,9858 1,0850 1,8195,75,6703 3,1369 3,4564 4,1186 4,931 1,0707 1,7996,487,6430 3,1055 3,41 4,078 4,8841 1,0577 1,7816,73,6185 3,077 3,391 4,0419 4,8411 1,0459 1,7653,078,5963 3,0516 3,3633 4,0090 4,801 0,9996 1,7016,133,5100,954 3,551 3,880 4,6516 0,9668 1,6571,0799,450,8838 3,1804 3,7943 4,5478 0,941 1,639,0408,4058,839 3,149 3,794 4,4710 0,97 1,5979,0103,3711,793 3,0818 3,6789 4,4114 0,9068 1,5769 1,9857,343,7613 3,0471 3,6383 4,3635 0,8937 1,5595 1,9653,301,7349 3,0184 3,6048 4,339 0,878 1,531 1,9333,839,6936,9735 3,553 4,60 0,8568 1,5113 1,9091,565,663,9396 3,517 4,154 0,8441 1,4948 1,8899,349,6377,918 3,4816 4,1786 0,8337 1,4813 1,8743,173,6177,8911 3,456 4,1488 0,850 1,4701 1,8613,06,6010,8730 3,4351 4,139 0,8176 1,4605 1,850,1901,5868,8576 3,417 4,108 0,8111 1,453 1,8406,1793,5745,8443 3,4017 4,0846 0,8055 1,4450 1,833,1699,5638,837 3,388 4,0687 0,8004 1,4386 1,849,1615,5543,84 3,376 4,0547 0,7959 1,439 1,818,1541,5459,813 3,3656 4,041 0,7919 1,477 1,813,1474,5383,8050 3,3560 4,0309 0,788 1,431 1,8069,1413,5314,7976 3,3474 4,007 0,7849 1,4188 1,800,1358,551,7908 3,3395 4,0114 0,7818 1,4149 1,7975,1308,5194,7846 3,333 4,

14 I. Jaiga, I. Garaj ,7790 1,4113 1,7934,161,5141,7789 3,356 3,995 0,7675 1,3969 1,7767,1074,4930,7559 3,990 3,9638 0,7591 1,3863 1,7646,0938,4775,739 3,796 3,9411 0,757 1,378 1,755,0833,4657,764 3,647 3,936 0,7475 1,3717 1,7478,0750,456,7161 3,58 3,9096 0,743 1,3663 1,7416,0681,4484,7077 3,430 3,8981 0,7396 1,3618 1,7365,063,4418,7006 3,348 3,8884 0,7338 1,3546 1,78,0530,4314,6893 3,16 3,8730 0,793 1,3490 1,718,0459,433,6805 3,115 3,8611 0,757 1,3445 1,7167,0401,4168,6735 3,033 3,8516 0,77 1,3408 1,714,0354,4114,6677 3,1966 3,8437 0,70 1,3377 1,7089,0314,4069,669 3,1910 3,8371 0,7117 1,37 1,6968,0180,3918,6465 3,170 3,8148 0,7067 1,310 1,6897,0100,388,6368 3,1608 3,8017 0,7007 1,3136 1,6814,0007,373,654 3,1476 3,786 0,697 1,3093 1,6764 1,995,3660,6187 3,1398 3,7770 0,6948 1,3063 1,6731 1,9914,3618,6141 3,1345 3,7708 0,6888 1,990 1,6647 1,981,3513,608 3,114 3,7555 0,6745 1,816 1,6449 1,9600,364,5759 3,0903 3,