Afschatting van verdelingen van kwadratische vormen met toepassingen op aanpassingstoetsen voor normaliteit en exponentialiteit

Summary In this paper a simple approximation is given for the distribution of the quadratic form being a weighted sum of squares of independent, identically distributed standardized normal variates. Using the formulae, the Monte Carlo results concerning some goodness of fit tests for normality and exponentiality are verified and extended.     

Maxima voor de eenzijdige overschrijdingskans en voor de grootte van de gemiddelde overschrijding bij discontinue en continue, eentoppige verdelingen

The well-known inequality of Bienaymé-Tschébyschef (for short B-T), generalized by Camp and Meidell (for short C-M) for continuous, unimodal distributions gives specific limits for total probabilities outside the ± to limits.
In many cases however, especially in the field of industrial applications we are interested only in the probability of one tail of the distribution, which of course must be smaller than the limits given by the B-T and C-M formula.
For these cases the maximum probability of surpassing the to limit on one side equals under B-T conditions and under C-M conditions instead of the two-sided values of 1/t² and 47/9 · 1/t² respectively (cf e.g. Uspensky: "Introduction to mathematical probability", 1937, p. 198) These results set upper limits for the value of

Alternatively we may also set an upper limit for the integral

which measures in terms of σ the average amount by which the limit + tσ is exceeded. This problem is also discussed and under C-M conditions an upper limit

is derived.
Some practical applications of these results are considered.     

Dimensionering van homogeniseringsinstallaties

Designing of homogenizing equipment
The spread in quality in a large stream of granular material can be sufficiently reduced only by homogenization in mixing heaps and bunker installations. In this treatment each unit of consumption is composed of portions which, at given intervals, are taken from the stream.
The paper indicates how to choose the number and size of the portions and the intervals between them in order to achieve the required reduction in spread at minimum costs.
This determination, elucidated by means of an example, is based on the correlo-gram of the stream referred to thin slices.     

De oplossing van een kwaliteitsprobleem door het gebruik van proefschema's

Samenvatting  Het artikel geeft een beschrijving van de strategie en de methoden, die worden gebruikt voor het zoeken naar de oorzaak van een kwaliteitsvermindering bij dubbelspiralen voor gloeilampen. Er wordt niet alleen aandacht besteed aan de beschrijving en motivering van de gekozen proefschema's maar ook aan de praktische uitvoering van de experimenten. Het fabricage-proces is nl. dermate gecompliceerd dat niet kon worden volstaan met de instructie de experimenten in aselecte volgorde uit te voeren.
Het gehele onderzoek wordt uitqevoerd aan de hand van een drietal proefschema's. De keuze van het tweede en derde proefschema volgt logisch uit de uitkomsten van de voorafgaande proef. Na vol-tooiing van de derde proef is ondubbelzinnig aangetoond dat de belangrijkste oorzaak van de kwaliteitsvermindering is gevonden.     

De bewerking van veldproeven volgens de methode van Knut Vik

Summary  (The treatment of field experiments by Knut Vik's method).
Agriculturists and statisticians have studied the problem, how to free the yields in a field experiment from the inequality of the soil. The method of the analysis of variance, due to the English statistician R. A. Fisher has taken an important place in the study of field experiments, so that other methods fell in the background.
The Norwegian agriculturist Knut Vik has suggested a method, which leads to good results without complicated calculations and without the knowledge of a great part of the mathematical statistics.
The mathematical foundation of this method is discussed in this article, which concludes, that the method of Knut Vik is of great importance for agriculturists, who don't want to perform intricate calculations.     

Assortimentsbepaling van seizoenprodukten

This paper deals with the analysis of a problem often met when determining purchasing policies. From a number of different kinds of goods, which are to be bought, the total quantity is restricted by storing capacity or financial limits. The problem "how much of each kind will be bought" is solved for a discrete as well as for a continuous distribution of demand for the various goods.     

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.     

Bepaling van stuwende en verzorgende bedrijfstakken door middel van statistische methoden

Determination of basic and non-basic enterprises by means of statistical methods.
In this article a sketch is given of the various methods used to obtain a clear insight into the meaning of the non-basic enterprises and the enterprises which because of their basic character belong to the factors determining the development of the population. Special attention has been paid to the method which puts the whole problem of the definition of basic and non-basic industries in the framework of a system of equations. This system has been built up from regional social accounts that have been developed for Amsterdam.     

Aantekeningen bij de herziening van de methodiek der prijsindexcijfers van het levensonderhoud

Momentarily (fall 196o) the Netherlands Central Bureau of Statistics is revising its price indexnumbers of family living. Some features of this revision are summarized below.
The old series is based on an expenditure pattern of 1951, whereas the new series will be calculated according to an expenditure pattern of 1959/'60. The latter data will be derived from a budget survey held among 250 households of manual and clerical workers consisting of 4 persons and grossing between four and eight thousand guilders a year (para. 6). The period covered was April 1959 till April i960.
The author indicates the way in which the varieties of the budget items to be covered by the monthly price surveys are chosen (para. 7). He discusses the principles and results of determining the number of price quotations (para. 8).
The choice of the municipalities in which price data will be collected is explained. An outline is given of the organisation of the new surveys apparatus (para. 14).     

Superpositie van twee frequentieverdelingen

Summary  (Superposition of two frequency distributions)
Notation:
n: number of observations
M: arithmetic mean
: standard deviation
μ_r: r^th moment coefficient
β₁: coefficient of skewness
β₂: coefficient of kurtosis.
The suffixes a and b apply to the component distributions. The suffix t applies to the resulting distributions.

The problem: Given the first r moments of two frequency distributions (to begin with μ₀). Find the first r moments of the distribution resulting from superposition of the two components ( r ≥ 5 ).
Formulae [1]. … [ 5 ] (§ 3 ) give the results in their most general form up to μ₄.
Some special cases are treated in § 4, and eight different cases of superposition of two normal distributions in § 5.
In § 6 some remarks are made about the reverse situation, i.e. the splitting into two normal components of a combined frequency distribution.     

Statistische analyse van muziek

This paper presents an application of statistical methods in analysing music. By counting the notes in pieces of music, frequency distributions are obtained and analysed.
Particularly Bach's compositions are considered, which appear to be rather disciplined as to the use of the 12 notes in dependence of the key. Perhaps it would be possible to characterize different composers by means of these frequency distributions.
Also a specimen of modern music without any key is discussed.     

Practische [berekening van de standaardafwijking van de constante term bij regressie-analyse

The formula arrived at in a previous article [1] is adapted to the usual computation schemes of regression analysis.     

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.