De invloed van de keuze der gewichten op de hoogte van een gewogen indexcijfer

The influence of the choice of the weights on the value of an indexnumber.
Price and quantity indexnumbers are weighted averages of groups of price and quantity ratios and they are convenient instruments to indicate the general tendency of such groups, especially if the number of basic ratios is considerable. The frequent use of indexnumbers is due to the fact that they can often be applied to problems for which, strictly speaking, an indexnumber had to be used derived from the same group of ratios but based on a different set of weights.
Two typical examples of such problems are given.
The use of a set of weights differing from the appropriate one is only justified, however, when the indexnumber is rather insensitive to changes in the set of weights. A simple formula is derived showing that the relative change of an index-number due to a change in the set of weights is equal to the product of the (weighted) coefficient of variation of the basic ratios, the (weighted) standard deviation of the relative changes of the weights and the (weighted) coefficient of correlation of the ratios and of the relative changes. The system of weights used in the calculation of these three factors is the same and is equal to the set of true weights belonging to the problem under consideration.
The practical use of the formula is demonstrated at the problem of index-numbers of costs frequently encountered in the practice of cost accounting.     

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 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).     

De vervaardiging van een systeem functie-ponskaarten

While making a series of 10,000 "function" -punched cards with the data of the well-known Barlow's tables and with the logarithms of the integers from the tables of Schrön, we took the opportunity to examine the accuracy with which our personnel and machines are working under normal conditions. We examined for instance in how many cases the first punching operator did not notice an error she made and how many of those errors remained after control-punching. These last-mentioned errors were traced by control-calculations with the punched cards, e.g. by calculating and checking n × n²= n ³ or by the calculation of differences. In this way it appeared that the first punching operator did not notice 270 errors in the part coming from Barlow's tables, being 0.055% of the number of punched columns or 2,7% of the number of cards. Further it was found, that only one error of the first operator was not noticed (or not corrected) by the controling operator.
Besides this a number of printing-errors, or anyway errors which were the result of the bad legibility of the digits, were found in Barlow's tables, where the punching operator and the controling operator had come to the same interpretation.
The article gives in some detail what precautions were taken by wiring the machines to attain that an error would be noticed as such.     

Poisson-proces met interrupties, een waarschijnlijkheids-model voor het machinaal dunnen van bieten.

Starting from the assumption that {even in case of dense precision sowing), the occurence of beet plants in a row can be taken as a realisation of a Poisson proces, the influence of mechanical thinning is studied.
The distribution of the distance between two plants, originally exponential, is changed by mechanical thinning in a way, dependent on the lengths of the intervals a (the potential survivor-zone) and b (the length of the thinningblade).
The distribution of the distance between plants after mechanical thinning, is examined, and the "best" way of thinning, leading to the distribution of the distance between plants with the smallest variance, is derived.     

Statistische eigenschappen van een vlambeveiliging met behulp van een gasontladingsbuis welke gevoelig is voor ultraviolet licht

Summary The output voltage of a detector circuit containing a gas-filled tube sensitive to ultraviolet radiation can be used to indicate whether a flame is still burning or not. As long as the flame is burning the random nature of the discharges of the tube leads to random fluctuations in the output voltage, which can be considered as a Markov-process with dependent increments. The problem is to select an alarm level for the output voltage with a suitable chosen probability of exceedance while the flame is burning, in order to limit the probability of false alarm. In this article a method is given for determining upper and lower bounds for the distribution function of the output voltage at any moment, answering the question of the alarm level to be chosen. Results are given for a special circuit.     

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.     

De spreiding van inschrijvingsbedragen op openbare aanbestedingen

Summary
The Dispersion of Bids in Public Contracting
The analysis is concerned with the coefficients of variation (standard deviation divided by mean) of bids in public contracting by the municipal authorities of a big city in the period 1946–1953. Three factors are investigated as to their impact on these coefficients of variation. The first is time; it is shown that the dispersion decreased gradually, probably as a result of increased cartelization. The second is the nature of the object (road building and earth-work, waterworks, and sewerage); waterworks seem to be characterized by a smaller dispersion of bids. The third is the size of the object, and it is shown that the dispersion is smaller for large objects than for small ones, probably as a result of the greater care of cost calculation for large objects. An attempt is also made to ascertain the interaction of these factors.     

De fout bij het gebruik van een tweetal benaderingsformules

Summary For the sake of efficiency, the use of simplified formulae in item analysis is recommended. For adequate use of such formulae the systematic error must be known. For two important formulae this error is estimated: the KUDER-RICHARD-SON-21 (instead of 20) and a formula for the correction of the spurious part-whole correlation. As for the KR, if itemscores are 1 andO, the KR20 is given in (I) and the KR21 in (2). The difference dr between them, given in (5), is directly proportional to the variance of the itemdifficulties. This variance turns out to be rarely greater than 0,05. If the distribution of itemdifficulties is unimodal, a value greater than 0,05 is indeed improbable in four-choice tests. The KR21, which does not need an item analysis, can be used as a substitute of the KR20. The KR21 coefficient itself is an lower limit, the KR21 plus dr - estimated by (6) - a probable upper limit of the KR20 coefficient. The part-whole correlation of tests is spurious. If one wants to correct for this spuriousness it is theoretically preferable to estimate the part-parallel test correlation instead ofthe part rest correlation. This is done here with the assumptions of classical test theory. The part-parallel test correlation is given in (12), the easily computed first approximation, called A, in (13) and the second approximation in (75), in which last formula the corrected correlation is expressed as a function of A, the standard deviations of part and whole, and the reliability of the test. One can compute with this formula a maximum A, below which the first approximation is sufficiently precise.     

De omvang van de steekproef bij een enkelvoudig steekproefsysteem

Summary  (Sample size for a single sampling scheme).
The operating characteristic of a sampling scheme may be specified by the producers 1 in 20 risk point ( p ₁), at which the probability of rejecting a batch is 0.05, and the consumers 1 in 20 risk point ( p ₂) at which the probability of accepting a batch of that quality is also 0.05.
A nomogram is given (fig. 2) to determine for single sampling schemes and for given values of p₁ and p ₂ the necessary sample size ( n ) and the allowable number of defectives in the sample ( c ).
The nomogram may reversedly be used to determine the producers and consumers 1 in 20 risk points for a given single sampling scheme.
The curves in this nomogram were computed from a table of percentage points of the χ² distribution. For v > 30 Wilson and Hilferty's approximation to the χ² distribution was used.     

De statistische basis van de planning in een machinefabriek

The normal or jigsaw-puzzle method of planning cannot be applied to the production of an engine factory. One has to fall back upon a kind of statistical automatism, like in the planning of road traffic as opposed to the planning of railway traffic.
In order to do this kind of planning efficiently, it is necessary to know the statistical relationships between the degree of occupation (number of working hours divided by number of available hours) of the machines, the waiting time and the velocity of flow through the factory.
The statistical analysis of some of these quantities in an Amsterdam plant, manufacturing medium size diesel engines, showed that Erlang's formulas of waiting time in telephone traffic (with exponential distribution of holding times) are applicable.
These formulas are used to prove that the highest degree of occupation is not the best one from an economic point of view. Formulas and graphs are given for finding the optimum degree of occupation in engine factories and other works where the same conditions apply.     

Commentaar op "Lineaire regressie, ook bij gegevens in strijd met het model"

Samenvatting Er wordt een voorbeeld gegeven waarbij de regressie-analyse volgens A ndrews zoals beschreven in het artikel van V an M ontfort niet tot het gewenste resultaat leidt.     

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.     

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.     

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.