UNKNOWN MEAN OF SAMPLED UNIVERSE 



641 



equally likely, that is, A^ = 0, c = - 3, a = 0.^« Here the precision 

 constant's a priori distribution is decidedly exponential and we might 

 predict the large probable and 99.73 per cent errors in the observed 

 average which actually result. 



Case 1 in Table I and Fig. 1 presents the problem in its entirety 

 with the resultant errors tabulated as well as shown graphically. 



(b) The engineer's knowledge, however, in all probability, is not so 

 limited as in (a) above, at least regarding the precision constant 



O 2 4 6 8 10 12 14 16 18 20 



T 



Fig. 2 — -Errors of averages of samples of size n. 



I — -99.73 per cent Error. 



11—90.00 per cent Error. 



Ill— 80.00 per cent Error. 



IV — 50.00 per cent Error. 



Note: Abscissa: T = n + 2+c + N. 



Ordinate: / = The Product of the Error of the Average and the Square 

 Root of B. 



(or the standard deviation). He knows that extremely small values 



of the precision constant are less likely than larger ones, and to some 



extent we picture the transition from (a) to this impression in the 



Cases Nos. 2 and 3 which as before may be found completely portrayed 



1" The formula for P{m) resulting from a substitution of these constants in 

 equation (10') reproduces the result obtained by Drs. J. Neyman and E. S. Pearson 

 for all values of the a priori function W'{m, a) equally likely: Biometrika, Vol. XX.4, 

 Parts I and II, July 1928; "On the Use and Interpretation of Certain Test Criteria 

 for Purposes of Statistical Inference," page 196, equation XXXV. 



