UNKNOWN MEAN OF SAMPLED UNIVERSE 633 



parameters may assume, and (2) the actual observed value of the 

 studied characteristic in each item of the sample. The a priori 

 information may be meager, in some instances hardly more than the 

 limits between which the parameters must lie, and again, from past 

 experience a great deal may be known about the universe, such as 

 its general form of frequency distribution, the most likely value for 

 each of its parameters to take, and a general feeling that they will 

 not, except in rare cases, lie outside of certain well defined ranges 

 closely bordering their believed most likely values. When the a 

 priori knowledge is meager, more weight must be attached to the 

 results of the sample, but when considerable a priori information is 

 at hand relatively less reliance should be placed in the sample; and 

 in some rare cases it is conceivable that so much is known before 

 the drawings are made that a particular sample, especially if small, 

 would justifiably be disregarded entirely. 



The Sampling of Attributes on the a posteriori basis for both 

 infinite and finite universes has already been set forth in these pages 

 at considerable length.^ The theory of Sampling of Variables when 

 the samples are large becomes usually a matter of assuming that some 

 of the parameters of the sample are sufificiently close to those of the 

 universe that no sensible error will be made in assuming them to be 

 equal. In this case the a priori knowledge of the universe, unless 

 far more exact than is normally found in practice, would exercise 

 but a slight effect in the conclusions which might be drawn, and is 

 therefore quite often properly neglected. 



When, for one reason or another, some conclusions are demanded 

 after having taken a small sized sample, it cannot safely be assumed 

 that the sample itself adequately describes the universe, and what a 

 priori knowledge we have must, of necessity, play an important role 

 in the determination of any legitimate statements as to the constitution 

 of the universe. 



The purpose of this paper is to study in strict accordance with the 

 theory of probability the conclusions which may be drawn concerning 

 the true parameters of the unknown universe after a "sample of varia- 

 bles" of any size has been examined. 



The paper is divided into the following five sections: 



I. The general equation is given for the a posteriori probability 



1" Deviation of Random Samples from Average Conditions and Significance 

 to Traffic Men," by E. C. Molina and R. P. Crowell, January 1924. "Some General 

 Results of Elementary Sampling Theory for Engineering Use," by P. P. Coggins, 

 January 1928. This second paper is based on another by Mr. E. C. Molina presented 

 before the Statistical Section of the International Mathematical Congress, held at 

 Toronto in August 1924. 



