Head of Department: Professor M. D. Goldman STAFF
Full-time Officers
Professor MAURTCE DAVID GOLDMAN, Ph.D. (Berlin), M.A.
Lecturer, NICOL MILNE, M.A. (Glas.)
(1) Investigation into the Origin of the Tribe of Benjamin Worker: M. D. Goldman
The tribe of Benjamin, the 'youngest' of the twelve tribes, seems to have been autochthonous in Canaan. Benjamin is the only one 'born' on Canaanite soil. The first judges were of that tribe. So was the first king of Israel. I n Judges XIX-XXI a war of all tribes of Israel against Benjamin is mentioned. Was it a war of conquest waged by the invading tribes against an original inhabitant of the country?
Docs Genesis X L I V : i8ff. point to a struggle amongst the northern and southern tribes over the inclusion of Benjamin into one or other of the amphictyonic confederacies? The investigation proceeds along these lines and into consideration and evaluation of all passages in which Benjamin is mentioned in the Bible.
(2) The use of the Old Testament in the Qur'an and the New Testament
Worker: N. Milne
Examining the Qur'anic teachings.
Examining the New Testament teachings.
Abstracting those with Old Testament similarity in each and evaluating their dependance on Old Testament.
THESIS PASSED FOR HIGHER DEGREE
MASTER OF ARTS
THOMPSON, J. A. The Significance of Transjordan and Its People for the History of Israel
49
Head of Department: Professor M. H . Belz STAFF
Full-time Officers
Professor MAURTCE HENRY BELZ, Chevalier de la Legion d'Honneur, B.Sc. (Syd.), M.Sc. (Cantab.), M.A., F.A.S.A., F.I.S.I.
Senior Lecturers:
HERBERT ARON DAVID, B.Sc. (Syd.), Ph.D. (Lond.) RUPERT THOMAS LESLIE, B.Sc, M.A.
Senior Tutors:
FRANCIS EMERIC BINET, Dr.Med. (Budapest) CHISTOPHER ROBIN HEATHCOTE, B.A. (W.A.)
The general pattern of previous years has been followed i n 1956, with special attention to mathematical developments i n Statistics, to the application of new and also of standard techniques in further- ing the research work of members of other departments within the University as well as i n outside institutions, and to industrial research involving statistical procedures.
The mathematical developments have included some investiga- tions concerning 'order statistics', information theory and the theory of linear programming. On the consultative side, assistance has been provided over a wide range of subjects, some leading to the develop- ment of new mathematical methods of analysis. In the industrial field advice has been given regarding the technique of linear pro- gramming, the full-scale setting up of experimental designs i n large plants, and the handling of multivariate regression systems. Some of the more interesting of these applications are described below.
The feature noted last year, that of the growing interest of out- side firms in the application of statistical methods to their own pro- duction problems, has been greatly expanded this year. Emphasis on this feature was provided by the stimulating visits, during 1956, of Professors M . G. Kendall, of the London School of Economics, S. S. Wilks, of Princeton University and A. W. Tucker, also of Princeton University, and Sir Charles Goodeve, F.R.S., of the British Iron and Steel Research Association. I n addition to addressing students and graduates in formal public lectures and in informal seminar sessions, these gentlemen addressed several statistical societies and visited numerous factories and research organizations in and near Melbourne.
Several seminar sessions on Linear Programming and Operational Research were given in the Department during the year, to the Department of Defence Production, to the Commonwealth Depart- ment of Works and to the School of Business Administration. Two lectures on the application of statistical methods in Traffic Research were also given by the Professor of Statistics.
SO
S T A T I S T I C S 51
Important firms visited during the year included the Broken H i l l Associated Smelters at Port Pirie, South Australia; the Australian Paper Manufacturers' plants at Maryvale and Fairfield; the Shell and Standard-Vacuum Oil Refineries; the Gas and Fuel Corporation of Melbourne, at West Melbourne, Highett and Morwell; the Aus- tralian Gas Light Co. of Sydney; the Petroleum and Chemical Corporation of Australia at Sydney; the Small Arms Factory of the Department of Supply; and the aircraft factory of the Department of Defence Production.
Details of the Report arc given under two headings, 'A, Theory of Probability and Mathematical Statistics', and 'B, Application of Statistical Methods'.
A. THEORY OF PROBABILITY A N D M A T H E M A T I C A L STATISTICS
(1) Introductory Text-book on Modern Statistical Methods Worker: M. H. Belz
Work on this project is continuing and approaches have been made from overseas regarding its publication.
(2) Estimation of Means of Normal Populations from Observed Minima
Worker: H. A. David
A paper with this title has been accepted for publication. It arose out of an enquiry, referred to in last year's report, by D. G. Ford (Aeronautical Research Laboratories) on the breaking strength of aeroplane wings.
(3) Project on Order Statistics Worker: H. A. David
A monograph on Order Statistics, representing the joint work of a group of statisticians, is to appear in the U.S.A. in 1958. I t will deal with both theory and applications. Chapters entitled 'Order statistics in short-cut tests' and 'Multiple decisions and multiple comparisons' have been contributed.
(4) Order Statistics in the Multivariate Case Workers: G. A. Watterson and H. A. David
Certain generalizations of the theory of order statistics to the case of multivariate normal distributions have been considered. In particular, the effect of ordering individuals with respect to one variable on the distribution of the remaining variables has been studied. This is of interest when individuals are selected on the basis of a certain criterion and one wishes to know the probable
results of this selection on other criteria correlated with the selec- tion criterion.
(5) Survey of Linear Programming Worker: Alison G. Doig
A survey of papers on linear programming and related topics is being carried out. The methods of extending the domain of validity of solutions are being investigated and particular atten- tion is also being given to developments of the technique to cover problems governed by non-linear relationships.
(6) Solution of 'Birth and Death' Equations Worker: C. R. Heathcote
Work is in progress in an attempt to generalize D. G. Kendall's solution of the 'Birth and Death' equations. Some results have been achieved in obtaining solutions for the case where the 'Birth' and 'Death' functions are polynomials in n, the size of the population, and arbitary in time t, but an analytic solution for the general case has not been found. As a result, an attempt is now being made to solve the equations by numerical methods.
(7) Counter Problem Worker: R. T. Leslie
In an attempt to provide a mathematical model for the function- ing of retinal receptors under very low illumination the following model was proposed. Light quanta arriving at the retina are known to follow a Poisson process. Suppose that the summation time of the receptor is such that a neural impulse is transmitted every time k quanta are received, provided these quanta arrive within a summation time. The problem is to obtain the distribution of impulse frequencies given the constant of thc Poisson process. This problem has proved difficult and a general solution has not been obtained although for small fe an exact solution has been found for the corresponding binomial process. However, the following analogous problem, in which the receptor is supposed to discharge whenever k quanta arrive, none of the successive intervals between quanta exceeding a specified time, has proved more amenable to analytic treatment and has been solved directly by considering sums of independent random variables, and indirectly by a limiting process from the solution of the corres- ponding binomial problem. The inversion of the Laplace transform of the density function has been carried out in collaboration with F. E. Binet.
(8) Information Approach to Disjunctive Response Times Worker: R. T. Leslie
Hick showed that the disjunctive reaction time for a key-tapping
S T A T I S T I C S 53
task was proportional to the information per symbol presented, and postulated a constant rate of processing of information. The follow- ing mechanism has been proposed as an explanation of the pheno- menon. Suppose the signal is repeated a number of times, as in fact occurs in discharges along afferent nerves. I f a 'potential' response occurs to each separate signal, a ' f u l l ' response is made when one particular potential response has occurred so often that the odds against any other being the required one exceed a subjectively set level. (Owing to 'noise' in thc system the same signal will not always evoke the same 'potential' response.) Repetition of the signal is thus postulated as the means employed in the afferent nervous system to introduce redundancy to the extent required to reduce the informa- tion per signal to channel capacity.
B. APPLICATION OF STATISTICAL METHODS
Computers: Betty Laby, Gwenda E. Jacobs and Johanna H.
Ronayne
(9) Linear Programming in Paper Manufacture
Referred by J. Wertheim (Australian Paper Manufacturers) Workers: M. H. Belz and Alison G. Doig
Paper is manufactured in wide rolls which are subsequently cut to the widths demanded by customers. The manufacturer's aim is to devise a cutting schedule (for a month's orders) such that the wastage arising from trimming the rolls to the required widths is a minimum.
Thc problem is amenable to solution by linear programming and several operating schedules were prepared by the computing staff of this Department.
(10) The Half-life of T h C"
Referred by D. L. Baulch and J. F. Duncan (Department of Chemistry)
Worker: H . A. David
• Over 250 series of observations, made at intervals of a minute, on thc decay of Thorium C" were available for the estimation of the half-life. A quality control approach was used to reject a number of series showing considerable irregularities. A joint note has been published.
(11) The Random Disorientation of Cubes
Referred by J. K. Mackenzie and M. J. Thomson (Division of Tribophysics, C.S.I.R.O.)
Worker: H. A. David
In this study interest centred on the probability distribution of
such quantities as the smallest of the nine angles between the prin- cipal axes of two randomly orientated cubic crystals. I t was pointed out that for small values of this angle an approximation to the distribution could be obtained; for example, the probability of a minimum angle of not more than 50 is approximately 0-034.
(12) Assays of Acetylcholine in Association with Certain Organic Complexes
Referred by A. Shulman (Department of Pharmacology) Worker: Alison G. Doig
Several series of six-point assays on the response of frog rectus muscle to acetylcholine associated with different metallic organic complexes were performed. The results of the assays, considered as a whole, formed a non-orthogonal experiment laid out in randomized blocks, with a possible covariance effect arising from different amounts of the complexes having been used in different assays. The analysis of variance indicated that there were genuine differences between the different metals used in the assays, but no dose-response effect within a particular metal could bc detected.
(13) Ionic Relations in Frog Muscle
Referred by F. H. Shaw and (Mrs) Shirley E. Simon (Department of Pharmacology)
Worker: R. T. Leslie
The results of several experiments on changes in the electrical activity of muscle cells resulting from changes in the ionic content of the environment have been analysed. The relationship between intracellular ions of Chlorine, Sodium and Potassium has been investigated and found not to conform to that predicted from current theories.