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

µ µ µ µn n n n1 2 1 2var( − ) = var( ) + var( ).1 2 1 2If the terms in the right side are known, we could have said bymeans of a table of the normal distribution whether the mentioneddifference can be explained by purely random causes or not. And forcalculating those variances (but not at all for applying the central limittheorem) we have to assume that the trials in both series areindependent, that is, that two series of Bernoulli trials were madewhereas the central limit theorem does not demand completeindependence. Then, if p 1 = p 2 = p (whose value is unknown),µ µ 1 1n n n n1 2var( ) + var( ) = p(1 − p)( + ).1 2 1 2It can be shown that the unknown p may be replaced here byµ + µˆp =n + n1 21 2and, assuming that the probabilities were identical, we see thatµ µ 1 1ˆ ˆn n n n1 2ξ = ( − ) ÷ p(1 − p)( + )1 2 1 2has an approximately standard normal distribution. Now, ξ can also becalculated only by issuing from numbers (µ 1 , n 1 ) and (µ 2 , n 2 ), i. e., bythe known results of experimenting. It is known that the absolutevalues of ξ exceeding 2 or 3 are unlikely. It follows that, whenobtaining values of that order, we should conclude that either thehypothesis p 1 = p 2 does not hold or, if it does, that an unlikely eventhad taken place.How to choose between these conclusions? Some authors think thatthe decision theory can allegedly numerically express the risk of eachpossible choice and thus help here. However, the risk there isexpressed by magnitudes which either are senseless or in any case willnever be known to the researcher. The theory based on a quantitativeexpression of risk is always useless except in studies of games ofchance 8 . Actually, the choice between the two mentioned decisions is acomplicated procedure; at present, it is impossible to study it withinthe limits of a mathematical theory.When solving such problems, we have to compare somehow theimportance of each possible solution should it occur wrong. Bothscientific and moral considerations denoted by the word conscienceare involved here. Approximately similar but somewhat simpler ischecking the hypothesis that the probability of success p in a givenseries of Bernoulli trials equals a given number p 0 . If it is also true fora series of n trials with µ successes, then36