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BELL SYSTEM TECHNICAL JOURNAL 



The errors found in Cases Nos. 8 to 19 on the various combinations 

 of a priori frequency curves He in a fairly narrow band distinctly 

 below those determined from the more conservative assumptions 

 underlying Cases Nos. 1, 2 and 3. This well illustrates the importance 

 of carefully surveying and as far as possible completely utilizing the 

 knowledge available before the sample has been made. 



{e) Finally, cases are bound to occur in which the engineer can 

 quite definitely say that some value of M other than x is a priori 

 most probable; this situation is encountered in Cases Nos. 20 and 21. 

 These are identical with Case No. 11 except that in Case No. 20, 

 M has been reduced about 6 ohms and in Case No. 21 raised about 



4! 42 43 44 45 46 47 48 49 50 51 52 53 



m = TRUE MEAN 



Fig. 5 — Typical a posteriori frequency distributions of the unknown mean. 



r / US'- \ f X — OT \n~(i/^H"+2+ei 



PW = ^"[1 + Bi^M - ^)^]-(™ [l + (-^)(^) ] 



2 ohms. The errors are somewhat increased by these changes in M, 

 as, of course, we should have predicted. Comparisons such as this 

 should help the investigator to decide whether or not his previously 

 selected figure for M is sufficiently close to x that they may safely 

 be equated. 



In the event that it is decided that M may not be set equal to x, 

 in any particular problem, as in Cases Nos. 20 and 21, the symmetrical 

 "Student" form of distribution for P{m), (except when iV = 0) no 

 longer occurs. This is clear from an inspection of Fig. 5 which 

 shows the three cases plotted on the same scale. 



It is suggested then, since the integral of P(m)dm here may become 

 difficult to handle, that recourse be had to the use of a planimeter 

 on the distribution plotted from equation (10) on rectangular co- 

 ordinate paper. In this way may be determined within what range, 



