1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

values of (2.5) are essentially less than those of the Kolmogorovstatistic (2.4). Therefore, when applying the Kolmogorov distributionfor (2.5), we will widen the boundaries of the confidence region andthus admit the hypothesis of normality more often than proper.And so, the careful practical application of the Kolmogorov test inthe most elementary (and therefore most common) situation isimpossible. That criterion helps in those cases when many suchexamples are available which were already tested by some statisticalcriteria and we wish to secure a general point of view concerning theirnumerous applications.Let us pass now to the essence of the problem on the confirmationof the Mendelian laws. Here is the classical situation. Some indicationhas two alleles, A (dominant) and a (recessive). Two pure lines withgenotypes AA and aa are taken and compulsorily crossed. A hybridwith genotype Aa emerges with its phenotype corresponding toindication A. Then a second generation is obtained under free crossing.When admitting the hypothesis of absolute randomness of thecombinations of the gametes, the probability of the occurrence ofgenotype aa is 1/4. Only individuals with genotype aa revealindication a in their phenotype so that the probability of its occurrenceis also 1/4. And so, if there will be n individuals in the secondgeneration, the number of occurrences of indication a in the phenotypemay be considered as the number of successes µ in n Bernoulli trialswith probability of success p = 1/4.This is the simplest case of the Mendelian law. Vast experimentalmaterial had been collected up to 1940 from which it was seen that inmany cases such a simplest law was indeed obeyed. Essentialdeviations (perhaps connected with a differing survivorship ofindividuals of different genotypes and other causes) was also revealed.The school of Lyssenko had been attempting to prove that that lawwas not working. To attain that aim, experiments were carried out, inparticular by Ermolaeva (1939). They were peculiar in that thematerial was considered not from all the individuals of the secondgeneration taken together, but separately for families. It is better toexplain the meaning of that term by an example. In experiments withtomatoes a family is consisting of all the plants of the secondgeneration grown in the same box. Each box is sown with seeds takenfrom the fruit of exactly one plant of the first generation. Theseparation into families occurs quite naturally.However, Kolmogorov (see above) showed that Ermolaeva’s mostnumerous series of experiments can be explained exactly by the mostelementary Mendel model. Suppose that for k families numbering n 1 ,n 1 , ..., n k the number of manifested recessive alleles was µ 1 , µ 2 , ..., µ k ,then the classical De Moivre – Laplace theorem [proving that thebinomial law tended to normality] leads to the normed magnitudes* µi− nip 1 3µi= , p = , q = 1− p =n pq 4 4ihaving approximately the standard normal distribution N(0, 1); theprecision of approximation is quite sufficient for n i of the order of102