THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING STATISTICAL ANALYSIS OF FATIGUE LIMITS USING THE LOGISTIC FUNCTION Ao Ho Soni Ro Eo Little October, 1963 IP-634

ACKNOWLEDGEMENTS The authors would like to thank Dr. Joseph Berkson for his helpful comments and his permission to reproduce the logits and antilogits tables. We would also like to thank Mr. Foster B. Stulen for providing ancillary information regarding the test data and Dr. William J. Schull for his criticism and advice.

TABLE OF CONTENTS Page ACKNOWLEDGMENTS. o.... e... o o o....... o. o o............. o o. o.. o o ii INTRODUCT IONo......... o o o........... o o.......... o...... o... o o Analogy Between Certain Biological and Fatigue Problemso 1 Logistic Function................................ 2 Estimation of Population Parameters.............ooo 3 Example of Estimating Population Parameterso........ 4 Standard Error of Estimates and Confidence Limito o. 5 Discussion.......................................... 6 REFERENCES 0 o.... o 0 o. o 0 o 0 o 0 e.. o o o o... o o o.0 o 0 0J 0 0 000 8 0 00 0 * 0. 8 iii

INTRODUCTION Statistical approaches which assume a life distribution pertaining to a given alternating stress amplitude have limited value when employed to estimate fatigue strengths associated with lives of 106 cycles or greater. For such lives, scatter of datum points is large and consequently numerical estimates are relatively weak. However, this weakness can be overcome by dealing with an alternating stress-percentage failed function. Moreover, this approach relates directly to the engineering concept of reliability, Development of the alternating stress-percentage failed function on the basis of fatigue data is not possible at present because insufficient data exist. However, it will be shown that the fatigue problem is analogous to problems successfully treated in the field of bio-statistics and that analysis of the limited fatigue data by bio-statistical techniques gives encouraging results~ Analogy Between Certain Biological and Fatigue Problems Suppose that in a biological study it is desired to estimate the effectiveness of a particular drug. Tests may be conducted in which N animals are given a dose, dl, and nl animals die, while N-n1 survive. Then, another group of N animals are given a dose, d2, and n2 -1

-2die, while N-n2 survive. These tests may be continued until many different groups have been subjected to some particular dosage. In this biological problem, the statistics recorded are the dosages and, at the end of a given period of time, the number dead and the number surviving. Analogously, in the fatigue problem, the statistics recorded are the alternating stress levels and, for a predetermined fatigue life, the number of specimens failed and the number that run-out. In other words, the alternating stress level in fatigue corresponds directly to the dosage in the biological problem. The logistic function has been used with success in the analysis of bio-assay problems such as the drug effectiveness analogy discussed above.(1-4) Consequently, because of the basic statistical similarities of these biological and fatigue problems, use of the logistic function is extended to analysis of fatigue limits as outlined below. Logistic Function This analysis consists of fitting fatigue data to the logistic function, which is P= -Q (1) 1 + e-( + Ps) (1) In Equation (1), P = the "true" percentage failed Q = the "true" percentage that run-out s = the alternating stress level a,d = population (not sample) parameters

-3The response is said to be quantal when it is measured not in terms of a continuous variable, but in terms of the observed number failed out of the total number tested at the given alternating stress level. In this analysis, it is assumed that the observed percentage failed, p, (based on a quantal observation) at the given alternating stress level, s, can be considered a random variable binomially distributed around the popu2 lation P at s with a variance a = PQN. Estimation of Population Parameters The logistic function is completely defined when both a and p are known; hence, the basic problem is to estimate these population parameters from experimental results. While several different techniques exist for estimating these parameters, the non-iterative Berkson(l) method simplifies the computational procedure and gives maximum likelihood estimates. Equation (1) can be transformed to give ln(P/Q) = a + Ps (2) Substituting a for ln(P/Q), Equation (2) becomes I = a + Ps (3) This linear transformation of Equation (1) is called the Berkson logit equation. The population parameters are estimated by a minimum X2 approach. Defining X2 = Z NiPiqi (hi -.i)2 (4) i=l

-4where k = the number of alternating stress levels = the estimated N[E = ln(^/A) = & + ~s] p = the observed percentage failed q = the observed percentage that run-out N = the number of specimens tested at the i-th stress level, the logit X2 is minimized with respect to the population parameters when Z Npq( - ~) =0 (5) Npqs(Q -') = 0. (6) Simultaneous solution of Equations (5) and (6) yields A^ w was - Z wi, ws (7) Z w Z ws2 - (Z ws)2 ea - w - Zws (8) w where w = Npq Whenever p = 0 or p = 1 is observed; p = 0 should be replaced by p = 1/2N, and p = 1 should be replaced by p = (2N - l)/2N. These "corrections" are required for a minimum variance of the estimates.(5) Example of Estimating Population Parameters The only data found in the literature which are amenable to analysis is that by Stulen.(6) Table I lists these data for a 4330 steel with an ultimate tensile strength of 130 ksi and a yield strength of 110 ksi.

-5The analysis of these data is presented in Table II. The logits used in the computational procedure are given in Table III. A Having computed the estimates O and p, p can be determined by using A A A a = a + Ps (9) 2 = ln(^p/) (10) and the antilogits of Table IV. The last two columns in Table II list 2 and p, respectively. Figure 1 shows the results of this analysis. It can be seen that this straight line estimate is somewhat conservative at each tail. It is clear that p is a function of the number of specimens tested and that p becomes critical at the tails when the sample size is relatively small. Unless "correction" factors are applied for p equal zero and p equal unity, these observations bias the estimates of the population parameters. Standard Error of Estimates and Confidence Limit The formulae for the variances of the estimates of the population parameters are a Zw Zw(s- / 2 1 S = - Z( s t)2 (11) where A = s/ s = Z ws/Z w

-6The scatter band for a desired level of confidence in the A estimated percentage failed, p, is given by j x, a ^ +E i - (13) +k2 X residual)[ + (s (13) k-2 A Y A(ss- 62 where residual - N(1-p)( )2 t = Student's t corresponding to the desired confidence level, with degrees of freedom = k - 2. For values of Student's t, refer to: R.A. Fisher and F. Yates, Statistical Tables for Biological, Argicultural and Medical Research, Fourth Edition, Oliver and Boyd, London, 1953. The 95 per cent confidence band appears in Figure 1. See Table V. Discussion It is very difficult to draw definite conclusions regarding the actual theoretical distribution from a single series of tests. While a visual comparison of plotted data and conceivable distributions may help in deciding which distribution is to be regarded as the most probable, it is generally not possible to exclude these other distributions on a theoretical basis.(7) In other words, in order that a certain distribution may be selected as the most probable and other conceivable distributions may be rejected by means of appropriate numerical analysis, it is necessary to have very extensive test data. Such data do not exist at present. As previously mentioned, the logistic function has found wide use in the field of bio-statistics. One important reason for its wide

-7use is that the population parameters can be estimated without becoming involved in iterative procedures. The computational procedures involved are adaptable to tabulation and are no more difficult than common least-squares curve fitting techniques. Yet, this method gives the maximum likelihood estimates of the population parameters. Simplicity of analysis and conservatism in analyzing the tails of the observed distribution are of prime importance in fatigue because of the many uncertainties involved in extrapolating these results to design applications. Although other approaches to analysis of fatigue limits have been suggested,(8) this method has definite advantages and should not be overlooked in fatigue analysis.

REFERENCES 1. J. Berkson, "A Statistically Precise and Relatively Simple Method of Estimating the Bio-Assay with Quantal Response, Based on the Logistic Function," Journal of the American Statistical Association, Volume 48, p. 565-599 (1953)o 2. F. J. Anscombe, "On Estimating Binomial Response Relations," Biometrika, Volume 43, p. 461-464 (1956). 3. J. Berkson, "Application of the Logistic Function to Bio-Assay," Journal of the American Statistical Association, Volume 39, P. 357-365 (1944). 4. Jo 0. Irwin and Eo A. Cheeseman, "On the Maximum Likelihood Method of Determining Dosage-Response Curves and Approximations to the Medium-Effective Dose, in Cases of a Quantal Response," Journal of the Royal Statistical Society, Supplement 6, p. 174-185 (1939)o See also E. Bo Wilson and Jo Worcester, "The Determination of L.Do 50 and Its Sampling Error in Bio-Assay," Proceedings of the National Academy of Sciences, Volume 29, p. 114-120 (1943). 5. Fo Jo Anscombe, "On Estimating Binomial Response Relations," Biometrika, Volume 43, p. 464 (1956). 6. F. B. Stulen, "On the Statistical Nature of Fatigue," Symposium on Statistical Aspects of Fatigue, American Society for Testing Materials Special Technical Publication No.121, p. 23-40 (1951), (See page 35 for data listed in Table 1). 7. Ao Io Johnson, "Strength, Safety and Economical Dimensions of Structures," Meddelanden NR 22, Statens Kommitte for Byggnadsforskning, Stockholm, Chapters 5 and 12, 1953, 80 A Tentative Guide for Fatigue Testing and the Statistical Analysis of Fatigue Data, American Society for Testing Materials Special Technical Publication No. 91-Ao (Supplement to Manual on Fatigue Testing, ASTM STP No 91), 19580 See also Do J. Finney, Probit Analysis, Cambridge University Press, London, 1952. -8

-999 98 "V__ i -+4 98 — _ — - - 97 - CALCULATED RESPONSE 97 /;/,, 96 ---- -- 95% CONFIDENCE BAND --- / -- 95 - +3 0 0 EXPERIMENTAL DATA / 90 I ------ 11"^~~~~~ _ __ It _ t F. Ru of i o R 1/nd F i Tests. +2 85 or cu 80 2o / /, 25 -— _- 30 20 -/ - _ / / / I I i i I II i I 0 — 2.0 — - - V7 i y - - o - - -2 — 4 — _'20_______ 3 I-I I i _ 1 -2 55 60 65 70 75 80 ALTERNATING STRESS LEVEL - KSI Figure 1. Results of Analysis of Rotating Bending Fatigue Tests. See Table I and Table V for analysis and Equation (13) for confidence band.

-10TABLE I RESULTS OF ROTATING BENDING FATIGUE TESTS ON SAE 4330 STEEL (Data by F. B. Stulen,(6) pertain to a fatigue life of 107 cycles) Stress Test Level Number Number Percentage Series ksi Tested Failed Failed 1 56 20 0 0O 2 58 23 1 4o3 3 60 22 1 4o5 4 62 21 2 9.5 64 20 1 5~0 6 66 20 3 15.0 7 68 27 12 44.4 8 70 27 14 519 9 72 20 17 85"0 10 74 20 20 100o0 11 76 18 18 100.0 12 78 24 24 100.0

-11TABLE II ANALYSIS OF ROTATING BENDING FATIGUE DATA 1 2 3 4 5 6 7 8 9 10 11 Si Ni Pi Test Stress Number Number Proportion 2 A A Series Level Tested Failed Failed Nipiti-pi) 1i siii i Pi 1 56.0 20 0 0.025*. 4875 -3.6636 -205.1616 3136.0 -4.7416.009 2 58.0 23 1 0.043 0.9465 -3.1026 -179.9508 3364.0 -4.0004.018 3 60.0 22 1 0.045 0.9455 -3.0551 -183.3060 3600.0 -3.2592 0.037 4 62.0 21 2 0.095 1.8055 -2.2541 -139.7542 3844.0 -2.5180 0.074 5 64.0 20 1 0.05 0.9500 -2.9444 -188.4416 4096.0 -1.7768 0.144 6 66.0 20 3 0.150 2.550 -1.7346 -114.4836 4356.0 -1.0356 0.261 7 68.0 27 12 0.444 6.6653 -0.2249 - 15.2932 4624.0 -0.2944 0.426 8 70.0 27 14 0.519 6.7403 0.0760 5.3200 4900.0 0.4468 0.611 9 72.0 20 17 0.850 2.550 1.7346 124.8912 5184.0 1.1880 0.767 10 74.0 20 20 0.975* 0.4875 3.6636 271.1064 5476.0 1.9292 0.873 11 76.0 18 18 0.972* 0.4899 3.5472 269.5872 5776.0 2.6704 0.935 12 78.0 24 24 0.979* 0.4934 3.8420 299.6760 6084.0 3.4116 0.968 Z ws = 1700.4220 Z w = 25.1114 Z wi = -10.0455 E ws2 = 115624.0432 Z vws = -502.4420 A w Z was - Z WI Z ws A wiZ A ws Zw ws2 - (Z ws)2 Zw Zw = 177.7904 = -0.4000 - 25.0952 479.7271 = 0.3706 = -25.4952 A a A i = a + Ps * Necessary corrections made for p = 0 and p = 1.

-12TABLE III LOGITS For p less than.50 on left, logit is negative. For p greater than.50 on right, logit is positive. Thousandtha, for p in left column 0 1 2 3 4 5 6 7 8 9.00 - 6.90675 6.21261 5.80614 5.51745 5.29330 5.10998 4.95482 4.82028 4.70149 4.59512.99.01 4.59512 4.49880 4.41078 4.32972 4.25460 4.18459 4.11904 4.05740 3.99922 3.94413 3.89182.98.02 3.89182 3.84201 3.79447 3.74899 3.70541 3.66356 3.62331 3.58455 3.54715 3.51103 3.47610.97.03 3.47610 3.44228 3.40950 3.37769 3.34680 3.31678 3.28757 3.25914 3.23143 3.20441 3.17805.96.04 3.17805 3.15232 3.12718 8.10260 3.07857 3.05505 3.03202 3.00947 2.98736 2.96569 2.9444.95.05 2.94444 2.92358 2.90311 2.88301 2.86326 2.84385 2.82477 2.80601 2.78756 2.76941 2.76154.94.06 2.75154 2.73394 2.71662 2.69955 2.68273 2.66616 2.64982 2.63371 2.61783 2.60215 2.5866.93.07 2.68669 2.57143 2.55637 2.54149 2.62681 2.51231 2.49798 2.48382 2.46984 2.46601 2.44235.92.06 2.44235 2.42884 2.41548 2.40227 2.38920 2.37627 2.36348 2.35083 2.33830 2.32591 2.31363.91.09 2.31363 2.30149 2.2946 2.27754 2.26574 2.25406 2.24248 2.23101 2.21965 2.20839 2.19722.90.10 2.19722 2.18616 2.17520 2.16433 2.15355 2.14286 2.13227 2.12176 2.11133 2.10100 2.09074.89.11 2.09074 2.08057 2.07047 2.0604 2.05052 2.04066 2.03087 2.02115 2.01151 2.00193 1.99243.88.12 1.99243 1.98299 1.97363 1.96432 1.95508 1.94591 1.9368 1.92775 1.91876 1.90983 1.90096.87.13 1.90096 1.89215 1.88339 1.87469 1.86605 1.85745 1.84892 1.84043 1.83200 1.82362 1.81529.86.14 1.81529 1.80701 1.79878 1.79059 1.78246 1.77437 1.76632 1.75833 1.75037 1.74247 1.73460.85.15 1.73460 1.72678 1.71900 1.71126 1.70357 1.69591 1.68830 1.68072 1.67318 1.66569 1.66823.84.16 1.65823 1.65081 1.64342 1.63607 1.62876 1.62149 1.61425 1.60704 1.59987 1.59273 1.58563.83.17 1.58563 1.57856 1.57152 1.56451 1.55754 1.55060 1.54369 1.53681 1.52996 1.52314 1.51635.82.18 1.51635 1.50959 1.50286 1.49615 1.48948 1.48283 1.47621 1.46962 1.46306 1.45652 1.45001.81.19 1.45001 1.44353 1.43707 1.43063 1.42423 1.41784 1.41148 1.40515 1.39884 1.39256 1.38269.80.20 1.38629 1.38006 1.37384 1.36765 1.36148 1.35533 1.34921 1.34310 1.33702 1.33096 1.32493.79.21 1.32493 1.31891 1.31291 1.30694 1.30098 1.29505 1.28913 1.28324 1.27736 1.27150 1.26567.78.22 1.26567 1.2598 1.25405 1.24827 1.24251 1.23676 1.23104 1.22533 1.21964 1.21397 1.20831.77.23 1.20831 1.20267 1.19705 1.19145 1.18586 1.18029 1.17474 1.16920 1.16368 1.15817 1.15268.76. 1.15268 1.14720 1.14175 1.13630 1.13087 1.12546 1.12006 1.11468 1.10931 1.10395 1.09861.75.25 1.09861 1.09329 1.08797 1.08268 1.07739 1.07212 1.06686 1.06162 1.05639 1.05117 1.04597.74.26 1.04597 1.04078 1.03560 1.03043 1.02528 1.02014 1.01501 1.00990 1.00479 0.99970 0.99462.73.27 0.99462 0.98955 0.98450 0.97945 0.97442 0.96940 0.96439 0.95939 0.95440 0.94943 0.94446.72.28 0.94446 0.93951 0.93456 0.92963 10.92471 0.91979 0.91489 0.91000 0.90512 0.90025 0.89538.71.29 0.89538 0.89053 0.88569 0.88086 0.87604 0.87122 0.86642 0.86162 0.85684 0.85206 0.84730.70.30 0.84730 0.84254 0.83779 0.83305 0.82832 0.82360 0.81889 0.81418 0.80949 0.80480 0.80012.89.31 0.80012 0.79545 0.79079 0.78613 0.78148 0.77685 0.77222 0.76759 0.76298 0.75837 0.75377.68.32 0.75377 0.74918 0.744603089 0.74002 02633 0.72179 0.7174 0.71271 0.70819.67.33 0.70819 0.70367 0.69915 0.69465 0.69015 0.68566 0.68117 0.67669 0.67222 0.66775 0.66329.66.34 0.66329 0.65884 0.65439 0.64995 0.64552 0.64109 i 0.63667 0.63225 0.62784 0.62344 0.61904.65.35 0.61904 0.61465 0.61026 0.60588 0.60150 0.59713 0.59277 0.58841 0.58406 0.57971 0.57536.64.36 0.57536 0.57103 0.56669 0.56237 0.55804 0.53373 0.54942 0.54511 0.54081 0.53651 0.53222.63.37 0.53222 0.52793 0.52365 0.5137 0.51509 0.51083 0.50656 0.50230 0.49805 0.49379 0.48955.62.38 0.48955 0.48531 0.48107 0.47683 0.47260 0.46838 1 0.46416 0.45994 0.45573 0.45152 0.44731.61.39 0.44731 0.44311 0.43891 0.43472 0.43053 0.42634 0.42216 0.41798 0.41381 0.40963 0.40547.60.40 0.40547 0.40130 0.39714 0.39298 0.38883 0.38467 0.38053 0.37638 0.37224 0.36810 0.36397.59.41 0.36397 0.35983 0.35570 0.35158 0.34745 0.34333 0.33922 0.33510 0.33099 0.32688 0.32277.58.42 0.32277 0.31867 0.3147 0.31047 0.3037 0.30228 0.29819 0.29410 0.29002 0.28593 0.28185.57.43 0.2815 0.27777 0.27370 0.962 0.2655 0.26148 0.25741 0.25735 0.24928 0.24522 0.24116.56.44 0.24116 0.23710 0.23305 0.22900 0.22494 0.22089 0.21685 0.21280 0.20875 0.20471 0.20067.55.45 0.20067 0.1 0.125 0.188 0.18452 0.180 0.1746 0.17243 0.16840 0.16417 0.16034.54.46 0.16034 0.15632 0.15229 0.14827 1 0.14425 0.14023 0.13621 0.13219 0.12818 0.1241 0.12014.53.47 0.12014 0.11613 0.11212 0.10811 1 0.10409 0.10008 0.09607 0.09206 0.08806 0.08405 0.08004.52.48 0.08004 0.07604 0.07203 0.06803 0.06402 0.06002 0.05601 0.05201 0.04801 0.04401 0.04001.51.49 0.04001 0.03600 0.03200 0.02800 0.02400 0.02000 0.0100 0.01200 0.00800 0..00400 0.00000.50 9 8 7 6 5 4 3 2 1 0 Thouaandths, for p in right column This table is reproduced with permission of Dr. Joseph Berkson. See Reference 1.

-13TABLE IV ANTILOGITS Entries give value of p for specified positive value of logit 1; if I is negative, p is 1 minus the tabled value. l 0 1 2 3 4 5 6 7 8 9 0.0.50000.50250.50500.50750.51000.51250.51500.51749.51999.52248 0.1.52498.52747.52996.53245.53494.53743.53991.54240.54488.54736 0.2.54983.55231.55478.55725.55971.56218.56464.56709.56955.57200 0.3.57444.57689.57932.58176.58419.58662.58904.59146.59387.59628 0.4.59869.60109.60348.60587.60826.61064.61301.61538.61775.62011 0.5.62246.62481.62715.62948.63181.63414.63645.63876.64107.64337 0.6.64566.64794.65022.65249.65475.65701.65926.66150.66374.66597 0.7.66819.67040.67261.67481.6770.67918.68135.68352.68568.68783 0.8.68997.69211.69424.69635.69847.70057.70266.70475.70682.70889 0.9.71095.71300.71504.71708.71910.72112.72312.72512.72711.72909 1.0.73106.73302.73497.73692.73885.74077.74269.74460.74e49.74838 1.1.75026.75213.75399.75584.75768.75951.76133.76315.76495.76C74 1.2.76852.77030.77206.77382.77556.77730.77903.78074.78245.78415 1.3.78583.78751.78918.79084.79249.79413.79576.79738.79899.80059 1.4.80218.80377.80534.80690.80845.81000.81153.81306.81457.81608 1.5.81757.81906.82054.82201.82346.82491.82635.82778.82920.83962 1.6.83202.83341.83480.83617.83753.83889.84024.84158.84290.84122 1.7.84553.84684.84813.81941.85069.85195.85321.83446.85570.85693 1.8.85815.85936.86057.86176.86295.86413.86530.86646.86761.86876 1.9.86989.87102.87214.87325.87435.87545.87653.87761.87868.87974 2.0.88080.88184.88288.88391.88493.88595.88695.88795.88894.88993 2.1.89090.89187.89283.89379.89473.89567.89660.89752.89844.89935 2.2.90025.90114.90203.90291.90378.90465.90551.90636.90721.90805 2.3.90888.90970.91052.91133.91214.91293.91373.91451.91529.91606 2.4.91683.91759.91834.91909.91983.92056.92129.92201.92273.92344 2.5.92414.92484.92553.92622.92690.92757.92824.92891.92956.93022 2.6.93086.93150.93214.93277.93339.93401.93462.93523.93584.93643 2.7.93703.93761.93820.93877.93935.93991.94048.94103.94159.94213 2.8.94268.94321.94375.94428.94480.94532.94583.94634.94685.94735 2.9.94785.94834.94883.94931.94979.95026.95073.95120.95166.95212 3.0.95257.95302.95347.95391.95435.95478.95521.95564.95606.95648 3.1.95689.95730.95771.95811.95851.95891.95930.95969.96007.96046 3.2.96083.96121.96158.96195.96231.96267.96303.96339.96374.96408 3.3.96443.96477.96511.96544.96578.96610.96643.96675.96707.96739 3.4.96770.96802.96832.96863.96893.96923.96953.96982.97011.97040 3.5.97069.97097.97125.97153.97180.97208.97235.97262.97288.97314 3.6.97340.97366.97392.97417.97442.97467.97491.97516.97540.97564 3.7.97587.97611.97634.97657.97680.97702.97725.97747.97769.97790 3.8.97812.97833.97854.97875.97896.97916.97937.97957.97977.97996 3.9.98016.98035.98054.98073.98092.98111.98129.98148.98166.98184 4.0.98201.98219.98236.98254.98271.98288.98304.98321.98337.98354 4.1.98370.98386.98402.98417.98433.98448.98463.98478.98493.9850 4.2.98523.98537.98551.98666.98580.98594.98607.98621.98635.98648 4.3.98661.98674.98687.98700.98713.98726.98738.98751.98763.98775 4.4.98787.9879.98811.98823.98834.98846.98857.98868.98879.8890 4.5.98901.98912.98923.98933.98944.98954.9965.98975.98985.98995 4.6.99005.99015.99024.99034.99043.99063.99062.99071.99081.99090 4.7.9909.99108.99116.99125.99134.99142.99151.99159.99167.99176 4.8.99184.99192.99200.9920.99215.99223.99231.99239.99246.99 4.9.99261.99268.99275.99283.99290.99297.99304.9910.9317.9924 0 1 2 S 4 1 5 6 7 9 This table is reproduced with permission of Dr. Joseph Berkson. See Reference 1.

TABLE V ANALYSIS FOR THE CONFIDENCE BAND 1 2 3 4 5 6 7 8 9 10 11 12 13 14 A wi A A Formula 13 Formula 13 A A 2_ ^ A (i. Ni Pi 1-p i i(i Ni (Pi- i) si si-s ( (-sign) Pi9 (+sign) 20.009.991.1784 -3.6636 -4.7416 1.16208 56 -12.6022 158.8154 -6.35969 -* -3.12351.04229 23.018.982.4065 -3.1026 -4.0004 0.80604 58 -10.6022 112.4066 -5.38923 -* -2.61157.06850 22.037.963.7839 -3.0551 -3.2592 0.04166 60 - 8.6022 73.9978 -4.41832.01189 -2.10008.10910 21.074.926 1.4390 -2.2541 -2.5180 0.06964 62 - 6.6022 43.5890 -3.47016.03018 -1.56584.17365 20.144.856 2.4653 -2.9444 -1.7768 1.36329 64 - 4.6022 21.1802 -2.72896.06123 -1.01934.26503 20.261.739 3.8576 -1.7346 -1.0356 0.48860 66 - 2.6022 6.7746 -1.63540.16247 -.43580.39174 27.426.574 6.6021 -0.2249 -0.2944 0.00483 68 - 0.6022 0.3626 -.80871.30789.21991.55478 27.611.389 6.4173 0.0760 0.4468 0.13749 70 1.3978 1.9538 -.09000.47752.98360.72711 20.767.233 3.5742 1.7346 1.1880 0.29877 72 3.3978 11.5450.53201.62948 1.84399.86296 20.873.127 2.2174 3.6636 1.9292 3.00814 74 5.3978 29.1362 1.09738.73302 2.76102.94048 18.935.065 1.0940 3.5472 2.6704 0.76878 76 7.3978 54.7274 1.63499.83617 3.70581.97611 24.968.032 0.7434 3.8420 3.4116 0.18524 78 9.3978 88.3186 2.15819.89660 4.66501.99071 A A 7 2 w = 29.7791 s = s = 68.6022 Z w(s-s)2 = 519.4373 Xresidual = 15545 Close to zero; beyon range of Table IV. * Close to zero; beyond range of Table IV.