Tables of Orthogonal Polynomial Values Extended to N: 104

Transcription

1 April, 9 Research Bulletin 97 Tables f Orthgnal Plynmial Values Extended t N: 0 By R. L. ANDERSON AND E. E. HOUSEMAN AGRICULTURAL EXPERI MENT STATION IOWA STATE COLLEGE OF AGRICULTURE AND MECHANIC ARTS STATISTICAL SECTION AGRICULTURAL MARKETING SERVICE UNITED STATES DEPARTMENT OF AGRICULTURE Cperating AMES, IOWA

2 CONTENTS Page Methds f calculatin and use 69 Intrductin 69 A cmparisn f available methds f calculatin. 696 The methd. 697 The fitting f plynmials by the e methd 00 Tabulatin f the e values 'li08 References cited '009 Tables 90

3 Tables f Orthgnal Plynmial Values Extended t n = 0 By R. L. ANDERS ON an~ E. E. HOUSEMAN METHODS OF CALCULATION AND USE INTRODUCTION Interest in rthgnal plynmials has been stimulated ill recent years, especially amng bilgists, by Fisher's use f them in evaluating a regressin integral 7), applicatin and extensin f which have fllwed each ther in rapid successin ) 9). Aside frm this specialized use, rthgnal plynmials are valuable in the study f time series bth in ecnmics and in agrnmy where ne may wish t study the effect f envirnmental cnditins n the yields f crps. If there has been a gradual depletin r imprvement in the level f fertility ver a perid f years, it shuld be evaluated and deviatins frm trend used in t.he study f envirnmental effects. Any reductin f the labr f calculatin ilvlved in such investigatins seems wrthy f sme effrt. If the independent variable is equally spaced, such as in time r space, the cnvenient methd f curve fitting by rthgnal plylmials can be used. The advantage ver the usual regressin methds f fitting nnrthgnal plynmials arises frm the fact that rthgnal plynmials are s cnstructed that any term f the plynmial is independent f any ther term. This prperty f independence permits ne t cmpute each regressin cefficient independently f the thers and als facilitates testing the significance f each cefficient. In additin the cmputing time fr curve fitting by means f rth I Prject f the Iwa Agricultural Experiment Statin., The authrs wish t acknwledge the material aid furnished by the Iwa State Research Cuncil making pssible the cmpletin f the cmputatins. They are indebted t Prfellsr G. w. Snedecr fr valuable criticism f the presentatin f the text. They wish t thank Prfessr R. A. Fisher.and Dr. F. Yates fr permitting the inclusin f the e values up t n=.

4 696 gnal plynmials is less, especially if a plynmial f degree greater than the secnd is fitted: The labr f fitting rthgnal plynmials t a lng series f data may be reduced thrugh the use f the frms intrduced by Yates. Designated' as e, and presented in a series f lectures at Iwa State Cllege in 97, t~ey have since been published by Fisher and Yates 8). The utility f this methd depends upn the availability f tables f e values f the same length as the series t be fitted. Since the existing tables are nly applicable t series f bservatins r less, it seemed wrthwhile t extend them in this bulletin t twice that number. With the generus permissin f the authrs and publishers f the riginal tables, we are able t present the cmplete set f e values up t and including thse fr n = 0. The 'bjectives f this bulletin are:. T present the tables f rthgnal plynmial values,. t illustrate the usel f the tables in fitting plynmials,. t give an elementary descriptin f the e methd,. t make a brief cmparisn f methds f fitting rthgnal plynmials and. t make a nte f hw the tables were extended and checked. A COMPARISON OF AVAILABLE METHODS OF CALCULATION The summatin methd, develped by Fisher 6), als presented by Snedecr ), is anther cnvenient methd f fitting rthgnal plynmials. In practice the e methd and the summatin methd are mst cmmnly used. Fisher and Yates 8) have made the statement that the e methd is decidedly mre expeditius than the summatin methd if nly a calculating machine nt a printing adding machine) is available, especially when the number f bservatins is small. Befre undertaking the task f extending the tables it was thught advisable t cmpare the tw methds rather extensively with respect t time f cmputing and accuracy when bth a calculatr and a printing adding machine are at hand, since the summatin methd invlves a cnsiderable amunt f cntinuus additin. We have cncluded frm varius cmparisns that:. If bth printing adding and calculating machines are available, the summatin methd requires almst 0 percent mre time n the average. This time is almst dubled when n adding

5 machine is available.. There is a tendency fr the relative efficiency f the e methd t be reduced slightly as the number f bservatins is increased.. The relative efficiency f the e methd is greater fr high degree plynmials.. If the investigatr is uncertain as t the degree f equatin suitable fr the regressin, he can save time in the use f the e methd, since new plynmial values are easily btained by adding t r subtracting frm) the ld values, while the summatin meth)d, which uses a prcess f differencing t btain the estimated values, necessitates recmputatin f the initial differences.. If n tests f significance f the reductin in sum f squares by succeeding terms f the plynmial is required, there may be little difference in the cmputing time f the tw methds. 6. Since there are fewer frmulas invlved in the e methd, it is learned much mre quickly by the cmputer.. If anyne intends training a cmputer t handle plynmials exclusively, we recmmend cnsideratin f a methd described by Aitken ). We fund the time f cmputatin culd be reduced belw that required fr the e methd, but the methd is t invlved fr rdina7 cmputing wrk. THE METHOD, At this pint a brief, simple descriptin f the e methd)s included. R. A. Fisher has presented a mre general discussin 6). Fr a mathematical presentatin, an article by F. E. Allan ) is cmmended. It can be prved that an rdinary plynmial [IJ always can be expressed in terms f rthgnal plynmials fr any specified set f values f x. Fr example, where x =,,..., 7, can be rewritten in the frm, 'i. e. plynmials which are independent nncrrelated).

6 li98 where the quantities in parenthesis are rthgnal plynmials. By definitin, tw plynmials, P and P', where x takes n a specified set f values, are said t be rthgnal if ~PP' = 0 As shwn belw the plynmials P l = + x), P = 8x + X), and P = 6 + 6,x ",:,;;' + ~ x ) satisfy the definitin f rthgnality:., P, p, p, PIP, PIP, p, P Ttal Fisher and Yates have designated plynmials such as Pl, P and P by the symbl e; ~: being a plynmial in x f the first degree, ~; a plynmial in x f the secnd degree, etc. Thus using their I!.tatin, equatin [] can be written in the general rthgnal frm, [] where the A "s are cnstants. Fr the imprtant case where x takes n the values,,,..., n, the relatinships between the e's and x up t the fifth degree are ~~=Al xx) ~~= A [ XX) n:~ ] ~~=A[ XX)LXX) ~~7] ~~= A[ XX)LXX) ~n;~ ) + n!~6 n 9)] /:"=' [ _) ) n 7»)+ ) n L 0n +07] ';; " x X X x 8 x x 008 where x is the mean value f x, and the A'S, which depend n n,.. In case x is a cntinuus variable the summatin is replac by integratin.

7 )99 are s chsen that the values f the ~"s crrespnding t the n values f x are integers reduced t lwest terms Since the ~"s are s cnstructed that ~~: ~; = 0 fr i ~ j, the terms f equatin [] are independent. The independence f the terms in [] ffers the previusly mentined advantages f this methd f curve fitting. Frm the abve table f values f the rthgnal plynmials PI, P and P a we see that the last half f the values f P are the same as the first half except that they appear in reverse rder. The same is true fr the values f PI and P a, but in additin t being in reverse rder the signs are als changed. The ~"s als have the same prperty; i. e., the sequences f values f and are symmetrical abut their middle terms.,,, The values f ~I' ~a' ~ are als symmetrical except that the values in the first half f the sequences are the negatives f thse in the latter half. See the tables f e values fr n < 9.) Since this prperty hlds fr all values f n, nly the lwer half f the e values when n is even and the lwer nt when n is dd are given in the tables, except fr n less than 9. In the fitting f plynmials f the frm [] the cnstants t be evaluated are the regressin cefficients, A~. They are I I I t I I chsen s that the quantity ~y A A ~... A k ~k), I I which is the sum f squares f the deviatins frm the regressin curve, is a minimum: After the A "s have been determined, ne can substitute fr the ~"s equivalent expressins in terms f x as given abve and thus change the regressin equatin frm the frm [] t the f '" =,,,, the crrespnding values f ~~=Aa[ xx)lx_x) n~~)j=a a [ x_x)a_:~x_x) ] Aa 9Aa 9A A arelo, 0' 0' 0' Hence A is taken equal t 0 s that t' = t;; ",.

8 ."00 frm []. Hwever, that is seldm necessary, since the estimated values f y crrespnding t the n equally spaced values f x can be cmputed, as shwn in the next sectin, by substituting the values f the ~"s in the regressin equatin f the rthgnal frm []. The sum f squares f deviatins frm regressin is given by the expressin ~ya~~y) A~~y~~) A~~y~~)... A~~y~~) The first telm, ~y, is the sum f squares f the riginal bservatins, and the secnd is the usual crrectin term fr the mean. The sum f the remaining terms cnstitutes the regressin sum f squares, the third term being the linear regressin sum f squa~es, the furth represents the reductin in the sum f squares f deviatins frm linear regressin btained by fitting a secnd degree plynmial, etc. THE FITTING OF POLYNOMIALS BY THE METHOD T illustrate the e methd f fitting rthgnal plynmials, we have chsen a series f 6 annual sugar prices 0), Frm the scatter diagram f the prices, fig., it appears that a third degree plynmial will fit the data nearly as well as well as a plynmial f higher degree. Thus in actual practice ne might be cntent t fit a plynmial f the third degree. Hwever, fr purpses f illustratin a furth degree curve is fitted in additin t a third degree curve. The directins fr cmputatin are presented in detail fr cnvenience f the cmputer. Part f the discussin is nt very pertinent s far as time series are cncerned, but may be helpful when ther kinds f data are used. The prices are in terms f mills cded by subtracting 0 mills frm each price; the cded figures are presented in table tgether with the e values fr n = 6, cpied frm page 60. The first clumn cntains the last half f the prices f them), beginning at the tp with the price fr 906 and ending with that fr 96. In the secnd clumn is the first half f the prices written in reverse rder; i. e., starting at the tp with the price fr 90 and ending with that fr 87. Thus ne can read the series in rder frm 87 t 96 by starting at the bttm f clumn, reading upward t the tp f the

9 TABLE. "0 CODED ANNUAL SUGAR PRICES, 8796, AND VALUES FOR N = 6. Price d I ~ ~ I ' M Sum clumn and then dwn clumn. If the number f prices had been dd, half f the median price wuld have been written at the tp f each clumn; i. e., if there were 6 prices and the thirtysecnd price was, the first item in each clumn wuld be 6. In clumn the numbers, s, are frmed by adding the crrespnding pairs f numbers in clumns and, + = 8 etc., while the numbers,, in clumn are frmed by subtracting the numbers in the secnd clumn frm thse in the first, = 8, etc. As a check, the sum f the third clumn, ~s, shuld equal the sum f the first tw clumns, and the sum f the furth clumn, ~d, shuld equal their difference; i.e., ~s=6=80+ and ~d= 70=80. The next step is the cmputatin f the regressin cefficlents. A' A' A" h. 0' "'" In t e equation,

10 ,"0 y= A~+A~~~+A~~~+A~~~+A~~~ The first, A I, is the mean, g, f the price series, A'=~s =,6 =8 N 6. The secnd cefficient A~ is simply related t the linear regressin cefficient, b, in the best fitting straight line Y = g + bx x). In fact A I is equal t b when n is dd, and A I equals 6 b/ when n is even. Here, A' ~dw 8))+)) )6) ~U~) 79,, 086 =79, =0., the sum f squares f ~/, ~~/), being taken frm the ft f I the clumn f ~/s fr n = 6. The remaining A "s are: A' ~s~~) ~ ~~) 7,77 = 70 7= 0.077,, A' = ~dw ~ ~~) 8)60)+)9) )0),70,7 _ 8) 96)+) 87) ),99) 9,8,,080, =98= ',,, A ' = ~sw = 8),96) ) 0,7) ~U~) 0,69,68,0 = 7,99, ,69,68,0 =. the quantities, ~)\ ~~/), and ~~/) als being taken frm the tables at the ft f the crrespnding clumns f ~"s. 6 ~~= xx) frddn. ~~=xx) frevenn.

11 ~0 If ne is fitting a plynmial f the third degree r less it may nt be wrth while t split the series int tw parts and cmpute sums and differences as in clumns and, table l. Instead cnsidering a secnd degree plynmial) the prices and e values can be arranged as fllws: Year Price ~I I ~ 90 ' i Thus AI = 6 = 8 6. A' = A' = 67) 6)+6) 9) ) 6) =0 79,.. 67) 0)+6) 7) ) 0) 0.077,70,7 We are nw at a pint where the tests f significance, table, TABLE. TESTS OF SIGNIFICANCE. Surce f variatin D egrees f freedm Sum f squares Mean square Ttal, l:y' Crrectin fr mean Deviatins frm mean Linear regressin Deviatins frm linear regressin Secnd degree term Deviatins frm quadratic regressin Third degree term Deviatins frm third degree regressin Furth degree term Deviatins frm furth degree regressin **Highly significant. *Significant ** 68** ** 9 7*

12 '0 f the varius regressin cefficients can cnveniently be made. The ttal uncrrected sum f squares f the prices, ':,y = ) + 8) ) = 08, and the crrespnding degrees f freedm, 6, are entered at the tp f the table. The crrectin term fr the mean, A I ':,s) =.8) 6) = 8,788, which takes ne degree f freedm, is subtracted frm the ttal sum f squares leaving,0 as the sum f squares f deviatins frm the mean. The sum f squares attributable t linear regressin, A' ':,d ~/) = 0.) 0,86) = 8, is subtracted frm,0 t give the sum f squares f deviatins frm linear regressin, 0,069. The value f F, 8 / =., shws that a highly significant prtin f the sum f squares,,0, is accunted fr by linear regressin. Of curse the significance f the linear regressin des nt necessarily mean that a straight line regressin gives a gd fit r that the regressin is linear. It means that the linear regressin line fits the data better than a hrizntal straight line drawn I I thrugh y. The sum f squares, A ':,s ~) = 0.077)7,77) = 68, assciated with the secnd degree term is the amunt which the residual sum f squares, 0,069, is reduced by fitting a quadratic equatin. The secnd degree term is highly significant since the value f F is 68/69 =.. Similarly the third and furth degree terms with the sums f squares A'':,d) = )087) = 886 and AI':,S~/) = )799,0) =7 are highly significant and significant, ref?pectively. Even thugh the furth degree term' is signi cant, it is nt very imprtant since the standard errr f estimate, V 9 =., wuld nly be reduced t V =.0 by fitting a furth degree curve instead f a third degree curve. The first, secnd and third degree terms all give ma"'ked decreases in the residual sums f squares. It may als be wrth pinting ut:. That the significance f any term higher than the first indicates nnlinear ppulatin regressin.. If the third degree term, fr example, is nt significant, nnsignificance f higher rder terms is nt guaranteed. Varius pssibilities can best be judged frm the scatter diagram. Fr the third degree curve the estimated values f Y, that is, the rdinates f the regressin curve, can be fund by substituting the values f the ~"s in the regressin equatin, [] Y =.0.~~+O.077 ~~O ~~

13 ~ 0 Thus in the first clumn f table we have the prduct f TABLE. CALCULATION OF THE ESTIMATED VALUES FOR THE THIRD AND FOURTH DEGREE POLYNOMIALS I I A /: " A' A' Estimated values fr rd degree plynmial A' t I Estimated values fr <; th degree plynmial II I I A = 0. and the values f ~, which were fund by put ting 0. in the calculatr and multiplying successively by I the values f ~: 0.)) 0. 0.)) )6) =.8 Similarly, the entries in clumn were btained by placing A I = in the calculatr and multiplying successively by th~ values f, and likewise fr clumns and 6. Estimated

14 'i06 values fr the third degree plynmial appear in clumns and in the same rder as the crrespnding prices in clumns and, table. Each entry in clumn is A I =. plus the sum f the crrespnding numbers in the first three clumns: = = =. In clumn each entry is. plus the sum f the crrespnding numbers in the first clumns with the signs f the numbers in clumns and changed: = = 8. The regressin equatin f the furth degree is Y=.0. { ~: ~ which differs frm equatin [] nly by the additin f the furth degree term. Since the rdinates fr the third degree curve have already been cmputed, the simplest way t find the rdinates fr the furth degree curve is t add the values f the term t the rdinates f the third degree I I curve. Thus in clumn 6 we have the values f A ~. In clumn 7 the entries are.9. = 9., etc., and in clumn 8,.9. = 8., etc., these nu'mbers being the rdinates fr the furth degree curve. If the subscript n had been dd, the numbers in clumn 6 wuld have been subtracted itlstead f added) frm thse in clumn t btain the entires in clumn 8, the reasn being the reversal f signs h the secnd half f the dd ~"s. Frm fig. we see that there is little difference between the third and furth degree curves, which was t be expecterl. Interplatins made frm the graph are accurate enugh fr mst practical purpses; hwever, since cases may arise where

15 ~ Price s were cded by subtracting 0 mills frm eaen price 80 \.., c ~ " 0 ~.,.!'! '" ci: 0.., Q) ; 0 ~ 0 87 \ 0, 0 0,, 0 0 th Degree ' \~ V:;=: \ 0 "'~ 0 ~.,\ " rdd.r.~ 0,. 0 ~_:_,:; "", 00 Qf , \ FiQure. Cded Sugar Prices, 8796 G. Watsn it is necessary r desirable t have t he regressin equatin in terms f x instead f e, we shall illustrate hw t change the abve third degree regressin equatin, [], int an equatin in terms f x. This is dne by substituting in [] the relatinships between x and the ~"s given in the previus sectin: Y =.0.AJ xx) A [ xx) n~ ] A [ xx) _ xx) en;~7)] Since n = 6, x =., and Al =, A = /, A = /, the A'S being taken frm t he ft f the clumns f the crrespndhg ~"s, we have Y=.0.) x.) ')[ x.)l0.] / ) [ x.) _ x.) 76.) ], which upn simplificatin becmes

16 08 l] Y = x.) x.) x.) Equatin [] culd be further simplified if desired by expanding the quantities in parenthesis. TABULATION OF THE f VALUES Tw methds f calculating the e values have been emplyed in rder t check the results. In the first methd the recursin frmula, was used, where r is the degree f the plynmial, in rder t find ~ values. The e values were then derived frm the ~ values using the relatinship e = A ~ where A is s chsen that the e values are integers reduced t lwest terms. The secnd methd utilized a prcess f differencing suggested by Aitken ). By using a printing adding machine with a subttal key the differencing can be carried ut easily fr plynmial values up t the fifth degree, but it is dubtful if this methd wuld be feasible fr higher degrees since the number f differences is equal t the degree f the plynmial calculated. Aitken used scalled T values which are related t the e values as fllws: e~t A', where A is the factr mentined abve fr changing the ~ values int e values. The quantity A' is the same fr each degree regardless f the size f n; its values fr the first five degrees are,, 0/, / and /0, respectively. Aitken als gave a frmula fr cmputing the sum f squares f the T values, which we have mdified t give the sum f squares f the e values. The mdified frmula, which is ~e= nr) nr+)... n+r) n+r) ~) " r r + ) r!) A; was used in cmputing the sum f squares f the e values.

17 'W09 REFERENCES CITED ) Aitken, A. C. On the graduatin f data by the rthgnal plynminals f least squares. Prc. Ry. Sc. Edinburgh, : ) Allan, F. E. The general frm f the rthgnal plynminals fr simple' series with prfs f their simple prpel'ties. Prc. Ry. Sc. Edinburgh, 0: ) Brandt, A. E. Tests f significance in reversal r switchback trials. Iwa Agr. Exp. Sta., Res. Bul ) Davis, F. E. and Pallesen, J. E. Effect f the amunt and distributin f rainfall and evapratin during the grwing seasn n yields f crn and spring wheat. Jur. Agric. Res., 0:. 90. ) Fisher, R. A. An examinatin f yields f dressed grain frm Bradbalk. Jur. Agric. Sci., : ) Fisher, R. A. Statistical methds fr research wrkers. Seventh Editin. Oliver and Byd, Edinburgh ) Fisher, R. A. The in'fluence f rainfall n the yield f wheat at Rthamsted. Phil. Trans. Ry. Sc. Lndn, Series B., : ) Fisher, R. A. and Yates, F. Statistical tables fr bi,lgical, agricultural and medical research. Oliver and Byd, Edinburgh ) Hpkins, J. W. Weather and wheat in western Canada. I. Influence f rainfall and temperature during the grwing seasn n plt yields. Canadian Jur. Res. : ) Schultz, Henry. Thery and measurement f demand. p University f Chicag Press. 98. ) Snedecr, G. W. Statistical methds. Cllegiate Press, Ames, Iwa. 90. ) Wishart, J. Grwthrate determinatin in nutritin studies with the bacn pig and their analysis. Bimetrika, 0:

18 TABLES OF e VALUES* n= n= n= n= I ~ ~ / /6 / / n=8 n=9 n= I I I I ~ ~ ~ ~ ~ / 7/ 7/0 /6 7/ /0 / *This prtin f the table is taken by cnsent frm "Statistical Tables" by l{. A. Fisher and F. Ya

19 ~I TABLES OF e VALUES*Cntinued) n= n= I I I ~ ~ ~ ~I ~I ~I I ~ / 7/ /0 /6 7 / 7/0 ~I I ~ n= n=6 ~I I ~ ~I ~I ~I ~I /6 / /0 0/ 7/ /0 *Thi. prtin 0 the table is taken by cnsent frm "Statistical Tables" by R. A. Fisber and F. Yates

20 TABLES OF e VALUES*Cntinued) n=8 n=9 I ~ t ' a 6 " , / / / /0 /6 7/ /0 n= n= ; /6 7/ /0 / / 7/ 7/ *This prtin f the table is taken by cnsent frm "Statistical Tables" by R. A. Fisher and F. Yate

21 TABLES OF e VALUES* Cntinued) n= n= I ~ / /. /0 /6 / /0 "This prtin f the tahle is taken hy cnsent frm " Statistical Tables" by R. A. Fisher and F. Ya t

22 TABLES OF e VALUES*Cntinued) n=7 n= ? /6 7/ /0 / 7/ 7/0 "Thi. prtin f the table is taken by cnsent frm "Statistical Tables" by R. A. Fisher and F. Yates, p

23 TABLES OF e VALUES*Cntinued) n=0 n= RO lo '; ) C ~ / / / /0 /6 / /60 0Thi. prtin f the table is taken by cnsent frm "Statitical Tables" by R. A. Fisber and F. Yates, PlI[,

24 TABLES OF e VALUES*Cntinued) n= n= I I ~ ~ ~ ~ t a ~ /6 7/ /0 / / 7/ 7/0 *This prtin f the tahle is taken hy cnsent Crm "Statistical Tahles"?y R. A. Fisher and F. Yates, pu

25 ~.....,...,=~~,~... TABLES OF e VALUES*Cntinued) n=6 n=7 I ~ ' l 6 Ii , / 7/ /0 /6 7/ /0 *This prtin f the table is taken by cnsent frm "Statistical Tables" by R. A. Fisher and F. Yates, pu

26 TABLES OF e VALUES*Cntinued) n=9 n= /6 / /0 0/ / /0.This prtin f the table is taken by cnsent frm "Statistical Tables" by R. A. Fisher and F. Yates, pub

27 ,... ~~.:...:.. ~ TABLES OF e VALUES*Cntinued) n= n= R / / 7/ /0 /6 7/ 7/0 *This prtin f the tabl~ is laken by cns~t frm "Statistic!>! Tab!e." by R. A. Fisher and F. Yates, publis

28 t' t' TABLES OF e VALUES*Cntinued) n= n= , /6 / /0 / / / /0 *This prtin f the table i. taken by cnsent frm "Statistical Tables" by R. A. Fisher and F. Yates, pu

29 TABLES OF e VALUES*Cntinued) n=8 n= !s R / 7/ /0,6 7/ 7/60 *This prtin f the table is taken by cnsent frm "Statistical Tables" by R. A. Fisher and F. Yates, pub

30 n= 9 ~ fi TABLES OF e VALUES*Cntinued) /6 7/ /0.This prtin f the table is taken by cnsent frm ' Statistical Tables" by R. A. Fisher and F. Yates, publ

31 ,0 TABLES OF e VALUESCntinued) n= ,9, ,9,870 / ,08, ,,867,80 /0 n= ,70,79,7,,99,80 9,86,87,060,9,0,6,60 / / 7/ /0

32 ,860 TABLES OF e VALUESCntinue<l) n= ,79, ,8,60 / ,96,988,70 / ,98,900,877,00 7/ n=6 7e ,0 :I,0,7 6,8,988,90 0/,9,799,8 7/,706,66,80,8 7/0

33 ' TABLES OF e VALUESCntinued) n= ' ,8 0,08, 6 9,0,6 8,6,97,8,0,09,,796 /6 7/ /0

34 "6 TABLES OF e VALUE8Cntinued) n=8 ~I ~I ~ ~I ,08 90, 87,,6 6,80,,6 9,097,79, 08 / / 7/ /0

35 ~7 TABLES OF ~, VALUESCntinued) n=9, ~ ' ,0,966,098 6,7,90,,9,8,90,776,9 /6 / /0

36 8 TABLES OF e V ALUE8Cntinued) n= Q ,980 8,86,0,06,,0 9,8,9,00,0,80,99,680 0/ /0

37 "9 TABLES OF VALUESCntinued) n=6 ~I ~I ~I ,90,68, ,0,0 89,06,899,66 8, 60,099, 86 /6 7/ /0

38 '0 TABLES OF e VALUESCntinued) n=6 ~I ~I ~I I ,,70,7 9,8, 0, 69, 68, 0,98,77,66,98 / / 7/ 7/0

39 TABLES OF t' VALUE8Cntinued) n= ,8 9,9,8 8,99,7 9,7,69,968 96,8,6,66,9 /6 7/ /0!

40 ' TABLES OF e VALUESCntinued) n=6 { ,60,97,9 7,9,96,60 7,98,8,6,066,69,80,896 0/ 7 7/0

41 _"B TABLES OF e VALUE8Cntinued) n=6 ~i { ; ls6,880 6,8,., 80,880,967,0,,60 09,0,08,80 /6 / /0

42 'i TABLES OF e V ALUE8Cntinued) n= ; ,80,66, 9,9,86,0,76,0,6,68,9,0,8 / / / /0

43 '; TABLES OF e VALUESCntinued) n= ,08 7,9, 9,98,76,,60,6,99, 96, 76 /6 / /0

44 ~6 TABLES OF e VALUESCntinued) n= ~ ,788 8,068,676,06,9,6 9,79,0,8 6,8,08,6 / 7/ /60 ~\

45 ~I ~I "7 TABLES OF e VALUESCntinued) n=69 { ~ ,70 78,9,,8,87,0 7,89,660,0 6, 8,0,9,8 /6 7/ /0

46 '8 TABLES OF e V ALUESCntinued) n=70 ~I ~I ~I ~I ,0,,9 8,6,7,060 7,76,690,,980 7,0.9,, 00 / / / 7/0

47 '9 TABLES OF e V ALUE8Cntinued) n= 7 ~ ' > _ ,80 0,0,6,9,7,80,667,7,090 0,090,,98,60 /6 7/ 7/0

48 '0 TABLES OF e VALUESCntinued) n= ' , ,80 0,0,6,9,7,80,667,7,090 0,090,,98,60 / 7/ /0

49 TABLES OF e VALUE8Cntinued) n= ,,06,60 09,09,70 9,8,,970,,0,07,00 /6 / /0

50 '' TABLES OF e VALUE8Cntinued) n=7 ~I ~I ~I ,0,079,0,0,0,700 0,0,86,0,6,997,98,70 / / / /0

51 TABLES OF e V ALUE8Cntinued) n= ,0 8,6, 890,0,77,60,06,80,96,0,67,8,770,700 /6 / /0

52 ' TABLES OF e V ALUESCntiDucd) n= ,00,07,060 7,97,07,00 6,0,08,00 9,9,909,7,960 0/ 7 7/60

53 " TABLES OF e VALUE8Cntinued) n= , ,08,0,00 8,8,80 70,8,96,80,7,70,07,80 /6 7/ 7/0 I

54 "6 TABLES OF e VALUE8Cntinued) n= ,8 6,060,0 69,,0 80,,86,0,067,7,9,,0 / / 7/ /0

55 'M7 TABLES OF e V ALUESCntinued) n=79 l ,080 7,08,06,7, 96,00 90,,96,0,6,,0,0 /6 7/ /0

56 8 TABLES OF e VALUE8Cntinued) n= ,60 8,90, 8,08,7, 60,909,69,880,9,,880,00 0/ / /0

57 '89 TABLES OF e VALUE8Cntlnued) n=8 ~I ~I ~I ' ' g \,80 7,06,76,66,707,0,7, 8,89,,80,6, /6 / /0

58 { '0 TABLES OF t V ALUESCntinud) n=8 / ~ Fi , ,76,,6 987,69,6,87,98,07, 6,00,00,80,90 / / 7/ /0 \

59 "a TABLES OF t' VALUE8Cntinued) n=8 { { ~ ,6,867,678 68, 660, 0,6,67,0,876 6, 00,,6,6 /6 7/ 7/0

60 ~ TABLES OF e V ALUESCntinued) n= , , " , ~ ,0 08, 97,8 6,867,9,0 99,9,, 68,00,79, 60,08, 0 / 7 / / 0

61 ' TABLES OF e VALUESCntinued) n= & , 70,6,8 7,9,,670,9,909, 0, 690 8,089,80,69,80 /6 / 7/0

62 ' TABLES OF e VALUE8Cntinued) n=86 ~ I ~ I ,990 6,9,9,.,860,977.07,080, ,8, / / 7/ /0

63 'J TABLES OF e VALUESCntinued) n=87. I ' : ,868 9,0,8 7,68,778,786,7,78 69,0,00,6, /6 / /0

64 "6 TABLES OF e V A LUESCntinued) n=88 ~ ' , : ~U ,8 9,99, 6,7, 9, 9,6,9,8,8,9,. 98 { {0

65 '7 TABLES OF e VALUE8Cntinued) n= ' ~ " ,70,00,97 0,90,69,60,69,0,,886 9,86,89,8,076 /6 7/ /0

66 "8 TABLES OF e VALUE8Cntinued) n= ; ,970 7,76,69 7,68,,0 7,7,09,86,60 9,677,06,67,7,60 / / / /0

67 Z9 TABLES OF e VALUESCntinued) n= , ,67, , 79, S, 90, 89,978, 9,88,9,90,88, 8 /6 7/ 7/0

68 ''60 TABLES OF e V ALUE8Cntinued) n=9 ~I ~I ~I ~I ~ , 6,9,0 8,80,067,76 907,80,89,66 696,79,096,7,8 / 7/ 7/0

69 '6 TABLES OF e VALUE8Cntinued) n= l ,0 7,6, 9,99,6,00,60,00,6 6,0,6,07,9 /6 7/ /0

70 'i6 TABLES OF e V ALUESCntinued) n= ,80 0,87, 6,,0,90 89,9,68,88 8,00,08,008, / / / /0

71 'i6 TABLES OF e VALUE8Cntinued) n=9 ~I ~I ~I ~I '; ' ,0,96,06 7,9, 06,0,7,90,0,90,8,09,7,00 /6 / /60

72 '6 TABLES OF e VALUE8Cntinued) n= , ~ ,880 07,6, 8 97,70,60,880,6,9,06, 0 87, 06,0,08,90 0/ 7/ /0

73 ~6 TABLES OF e VALUE8Cntinued) n=97 ~ ,08 7,68, ,78,00,87,9,67,60,7,068,87,,60 /6 7/ 7/0

74 ,66 TABLES OF e VALUE8Cntinued) n= ~ ,698,7,P0,9,9,60 6,,6,,80,8,698,7,07,760 / / 7/ 7/0.

75 '67 TABLES OF e VALUE8Cntinued) n=99 ~ ' ~ ' R ,80 7,,70,08,69, 00 7,07,88,6,080,,9,8,09,70 /6 7/ /0

76 TABLES OF e VALUESCntinued) n= , ,7, j ,70,0,700 8,080,76,,600,9,,,600 0 / / /0

77 '869 TABLES OF e V ALUE~Cntinued) n= lol fi,, C\' ;; ,80 8,60,80 6,,77,60 7,77,0, 70 9,98,6,6,70 /6 / /0

78 '>70 TABLES OF e V ALUEBCntinued) n=0 f f ~ \ ,70 7,9,780,,,70 87, 69,,060,9,0,68,86,060 / { / /0

79 ~7 TABLES OF e VALUESCntinued) n=0 f ~ I r ,0 6,7,76,8,0,90 0,09,9,0,0, 7,9,,660 /6 7/ /60

80 '7 TABLES OF f VALUESCntinued) n= )) ' ' _ , ,90 67,60,8,9, 0, 90 0,9,908,69,0,9,78,0,,880 0 / 7/ 7/0