Een noodzakelijke correctie van de overschrijdingsgrenzen voor steekproefgemiddelden bij toepassing van stratified sampling

Summary  (A necessary correction of the control chart limits for averages of samples in the case of stratified sampling).
Application of stratified sampling results in smaller fluctuations of sampling than where the same total number of individuals is drawn at random from the superposed strata.
The proportion of the standard errors of the averages obtained by these two sampling methods may be expressed by a factor φ (o ≤φ≤ 1). The probability limits, between which the random sampling results would be fluctuating normally, should be corrected according to this factor.
A few properties of φ are discussed. This is graphically illustrated. Remarks have been added about the relation between the shape of the population of the sampling averages and the population from which the individuals are drawn, and about the difficulties which arise when the populations are non-Gaussian.     

Een toepassing van “importance sampling”

Summary This paper describes an experiment with “importance sampling”, to show how much reduction of the computation time and sample size can be achieved in comparison with the usual Monte Carlo method. A comparison is made between each of the three methods of “importance sampling” and the usual Monte Carlo method by the determination of the expression Of the three methods A, B and C the first one uses the shifted exponential distribution, the second one uses the gamma distribution, and the third one uses the exponential distribution with modified parameter. These three methods have all smaller variances, ranges and sample sizes than the usual Monte Carlo method. Their order of preference is A, B, C. With respect to computing time only the method A is significantly better. So only the method A is an improvement in respect of both the sample size and the computing time.     

Een lading van kraaltjes

Summary An urn containing 200 red beads and 1000 yellow ones is frequently used for demonstration of the hypergeometric or binomial distribution. Experiments show that the practical frequency distribution differs slightly but definitely from the theoretical one. This deviation turned out to be caused by electro-static forces.     

Regen, een toepassing van dynamische programering

Veel mensen zal het wel eens overkomen zijn dat ze tijdens een wandeling overvallen werdcn door een regenbui. De eerste reactie in zo'n situatie is in de regel, hard te gaan lopen teneinde zo droog mogelijk thuis te komen. Bij sommigen echter zal ongetwijfeld wel eens de vraag zijn opgekomen hoeveel effect dat harde lopen nu heeft; als je hard loopt ben je weliswaar eerder thuis maar je vangt ook meer regen per tijdseenheid, en stel je voor dat het al gauw ophoudt met regenen. In dit verhaaltje wordt het probleem van de optimale strategie aan de orde gesteld, wat is de optimale snelheid in verschillende omstandigheden. Als criterium gebruiken we de (verwachte) totale hoeveelheid opgevangen regen. Daarbij gaan we er van uit dat de regen recht naar beneden valt en homogeen verdeeld is over de ruimte. In paragraaf 1 wordt aangenomen dat het blijft regenen. In paragraaf 2 wordt rekening gehouden met de mogelijkheid dat het ophoudt met regenen en daarna droog blijft. In paragraaf 3 nemen we aan dat het steeds opnieuw kan gaan regenen.     

Een systeem van kostenbeheersende productieplanning

In this paper a system of cost-controlled production planning is described. This system considers all kinds of costs associated with a production in phases (e.g. production of units, production of subassemblies, assembly) which may be affected by the planning, such as set-up costs, costs of transport, control, inventories, capacity and changes in capacity.
The mathematical model leads to a linear or mixed discrete-linear programming problem whose solution gives for each time period considered the size of the capacities which should be used and of the series of different products which should be produced. Practical recommendations are given for obtaining a sufficiently satisfying solution.     

Produktiviteitsverhoging bij accountantscontrole door toepassing van gelaagde steekproeven

The auditor's productivity: application of stratified sampling to an auditing problem.
In this paper it is shown that the use of stratified sampling can lead to a considerable saving of costs in auditing operations and thus increases the auditor's productivity.
The sampling method to be described will render it possible to say that the total value of the transactions will not differ more than a fraction φ, given in advance, from the correct total value, apart from a probability not surpassing a preassigned level e, on condition that a certain acceptance criterion is fulfilled. Taking into account the auditing costs, the author gives an optimal sampling design and calculates the savings obtained.
Finally the theoretical results are applied to a population of clerical transactions, with an exponential and a logarithmico-normal distribution respectively.     

Een steekproef ter vaststelling van de verdeling van de hoenderstapel

A sample estimate of the poultry distribution
In 1954 a sample survey was carried out of the returns of the agricultural census, May 1954, in order to establish the distribution of the number of holdings and the number of poultry, by size-group of arable land and by size-group of number of poultry. A simple 10 per cent, sample was planned and the expected variances of the numbers of poultry in each size-group were estimated beforehand. The method of estimation is indicated in this paper. This resulted in comparatively high relative errors in the size-groups above 500 hens and also in the size-groups of arable land. Therefore holdings with 500 hens and over were enumerated completely. Some tests were performed with the sample results and finally the efficiency of the stratification is discussed.     

Het oordeel van een bedrijfsleider over de toepassing van de statistiek in het bedrijf

A manager's view on the application of statistics in the concern.
To be able to judge of the correct dosing of statistical methods in his concern a manager should discriminate between incidental research and systemetical application of statistics. A sound job limitation and good information are indispensable to the success of a statistical check system.
The cost of the introduction of statistical check methods into a concern with a variated manufacturing programme in the field of electrical engineering (350 people) was estimated to be Djls 18,000 in one year. In the next year more than this amount will be earned back through the savings and the improvements obtained.     

Optimale rantsoenering van aardolie en aardolieprodukten in Nederland; een toepassing van input-outputanalyse en lineaire programmering

Samenvatting Bij een invoerrestrictie van ruwe olie ontstaat het probleem van de optimale verdeling van de olie over de verschillende bedrijfstakken. Met behulp van input-outputanalyse kan dit probleem worden getransformeerd in een lineair-prograrnmeringsvraagstuk. In het navolgende artikel worden een aantal oplossingen gepresenteerd die men verkrijgt bij verschillende veronderstellingen omtrent bezuinigingsmogelijkheden, verbruiksminima enz.     

Een criterium voor het kiezen van een proefopzet

A characterization of D-optimality is given together with several examples where D-optimal designs are computed.     

Enkele opmerkingesn over de mogelijkheid van toepassing van mathematisch-statistische methoden bij tijdstudieproblemen en in het bijzonder bij het Bedaux-systeem

Summary  (Some remarks on the possibility of applying mathematical-statistical methods to time-study problems and in particular to the Bedaux system.)
Time and motion study are based on observations of the workman's movements and an estimation of his efficiency.
The time spent on a certain action in mass-production is not constant but shows a considerable variation, due to various causes. The time-study-man who investigates this action and expresses it in a rate, causes still more variations by certain inaccuracies in using the stopwatch and calculating the rate itself. Considered from a statistical point of view, these cumulating variations result in a standard-error in the calculated rate; this gives rise to some interesting phenomena.
A more statistical approach tot the whole problem is shown in this article to be possible.
In the discussion, the application of the Beaux-system is assumed, but the conclusions hold good for all time study systems.     

Een Monte-Carlo-bepaling van overschrijdingskansen, in verband met een keuzetest

A Monte-Carlo method for a test of significance, applied to points on a lattice, in connection with a vocational preference test, by C. A. G, Nass.
Appendix by Constance van Eeden.
A periodical rectangular lattice, with a period of k.m, is considered. Thus there are N = k.m points on the lattice, repeated in the two perpendicular directions. Two points are said to be "connected" if they are adjacent in a straight or diagonal way. Thus, if k and m 3, every point is connected with 8 other pooints. Out of the N points of the lattice, n points are selected and the total number of connections x, of all possible pairs of those n points is considered for a vocational preference test with k = m = 9, N = 81, n = 10. The problem is to test whether the sum y = x₁+…+ x*** from a sample of h values of x, is significantly small, under the hypothesis that in the h cases the n points are selected at random with equal chance. A Monte-Carlo sample of 100 values of x was taken, using random numbers. For h = 1, the problem was solved by the determination of P( y x₁), assuming that y is taken at random from the 101 values of x, supplied by the Monte-Carlo sample and x₁ for fixed values of x₁. For h - 2, a similar solution is given. For greater values of h, Student's two-sample test, with correction for continuity is suggested. For h = 2 the results of Student's test are compared with those of the solution mentioned above.
In the appendix a summary is given of results found by P.A.P Moran and P. V. Krishna Iyer for some closely related problems. Further some results concerning exact distributions, moments and asymptotic distributions for C. A. G. Nass' problem are given. The proofs of these results may be found in a paper by C. van Eeden and A. R. Bloemena (1959).     

Efficient high-dimensional importance sampling

The paper describes a simple, generic and yet highly accurate efficient importance sampling (EIS) Monte Carlo (MC) procedure for the evaluation of high-dimensional numerical integrals. EIS is based upon a sequence of auxiliary weighted regressions which actually are linear under appropriate conditions. It can be used to evaluate likelihood functions and byproducts thereof, such as ML estimators, for models which depend upon unobservable variables. A dynamic stochastic volatility model and a logit panel data model with unobserved heterogeneity (random effects) in both dimensions are used to provide illustrations of EIS high numerical accuracy, even under small number of MC draws. MC simulations are used to characterize the finite sample numerical and statistical properties of EIS-based ML estimators.     

Een nomogram voor betrouwbaarheidsgrenzen van quantielen en overschrijdingskansen bij normaal verdeelde grootheden

A nomogram for confidence intervals and exceedance probabilities.
In this paper two problems are considered regarding the probability β that an observation on a normally (μ, σ²)-distributed random variable exceeds a given value W:

If μ and σ² are unknown, the two problems are as follows:
1)if Wis given, to determine a confidence interval for β and
2)if β is given, to determine a confidence interval for W.
For these two essentially equivalent problems graphs are given from which the confidence intervals can be determined. The graphs are given in terms of:

and are based on an approximation for the distribution of x¯ +k s .     

Een prijsindex van nieuwe personenauto's, 1950–1965*

Summary In constructing a price index of new motorcars the main problem is how to deal with the introduction of new models and with quality variation generally. The first difficulty is turned by compiling year-to-year indices based on models that each account for 2 per cent or more of total sales in either year. The shifting basis of these indices comprises from 7 to 15 models which together account for 45 to 70 per cent of total registrations. All models are subject to minor improvements; this quality variation is taken into account by representing increases in horse power and overall length by equivalent price reductions. The elasticity coefficients involved in this translation are derived from a cross section analysis of prices and technical characteristics of some fifty widely ranging models in 1964. A price index for the period from 1950 to 1965 is then obtained by linking successive year-to-year indices. Over the whole period list prices have risen by a third, to a large extent as the result of changes in purchase tax; this price rise is just about offset by imporvements in quality of some 2 per cent per annum.