XIIL.—SIMPLE SAMPLING OF ATTRIBUTES. 271 
The actual difference is 3-0 per cent., or over 5 times this, and 
could not have arisen through the chances of simple sampling. 
If we assume that the difference is a real one and calculate the 
standard error by equation (6), we arrive at the same value, viz. 
0-56 per cent. With such large samples the difference could not, 
accordingly, be obliterated by the fluctuations of simple sampling 
alone. 
Case III.—Two samples are drawn from distinct material or 
different universes, as in the last case, giving proportions of 
4A’s p, and p,, but in lieu of comparing the proportion p, with 
py it is compared with the proportion of 4’s in the two samples 
together, viz. p,, where, as before, 
2 TP tT np, 
OC mtn 
Required to find whether the difference between p, and p, can 
have arisen as a fluctuation of simple sampling, p, being the 
true proportion of 4’s in both samples. 
This case corresponds to the testing of an association which 
is indicated by a comparison of the proportion of 4’s amongst 
the B’s with the proportion of A4’s in the universe. The general 
treatment is similar to that of Case II., but the work is complicated 
owing to the fact that errors in p, and p, are not independent. 
If ¢, be the standard error of the difference between p, and 
Py We have at once 
a =6+6—2r,. qq 
1 1 1 
=pq r= rp ! 
Yn +m, on, Nn, +n, 
Ta being the correlation between errors of simple sampling in 
py and p,. But, from the above equation relating p, to Py 
and p,, writing it in terms of deviations in p, p, and 2 
multiplying by the deviation in p, and summing, we have, 
since errors in p, and p, are uncorrelated, 
2% q Ta 
rte nin 
17% 177% 
Therefore finally 
& = Pd ny 
Yimin @ 
Unless the difference between py and p, exceed, say, some 
three times this value of ¢, it may have arisen solely by the 
chances of simple sampling.