APPENDIX I 
3717 
the influence of gross earnings on net earnings is substantially 
the same for banks in any group, the conclusion is unavoidable 
that the influence of total expense on net earnings is significantly 
different for banks in the different groups. This fact is shown 
particularly in the mean v’s. Note that the mean v’s for Groups 1 
and 2 are opposite in sign and 4 times their own probable errors 
(+ 1.2 =0.3 and — 1.1 += 0.3). The mean v is the algebraic 
sum of the 4+ and — v’s divided by the total number of v’s or 
banks; for example, for Group 1 it is (324-173) = 129= 
+ 1.2. The probable error of the mean v for each group is com- 
puted from the relation r, ===, where r, = the probable error 
of the mean v, r = the probable error of a single observation of 
Solution /,, + 3.5, and # the number of banks in the group. The 
mean 2’s for Groups 3 and 4 are also opposite in sign, and four 
times their own probable errors. According to the laws of prob- 
ability, there are only about 7 chances in 1,000 that these mean 
v’s, four times their own probable errors, are entirely fictitious. 
This is the most significant evidence pointing to a different effect 
of total expense on net earnings for banks in the several groups. 
The interpretation of the signs of the mean residuals is as fol- 
lows: Referring to equation (1), in which are substituted the 
specific constants (5) from Solution I,, it is evident that if the 
observed decrease in net earnings, —AN, is larger than the com- 
puted increase, + 1.08 — 0.497 AT, the residual is 4, and vice 
versa. Hence, in Solution 7,, the observed decrease in net earn- 
ings was larger in Groups 1 and 3 than the increase computed 
from (6). In Groups 2 and 4, the observed decrease in net earn- 
ings was smaller than the computed increase. Bearing in mind 
the assumption of uniform influence of gross earnings on net 
earnings in all groups, this fact is to be interpreted as follows: 
in Groups 1 and 3, the influence of total expense on net earnings 
is probably larger, and in Groups 2 and 4, smaller, than that 
represented by (6), based upon all 408 banks. 
If it is now assumed that the influence of total expense on 
net earnings is the same in all groups, and if the preceding 
processes for the 408 banks, using equation (2) instead of (1), 
are repeated, there is obtained, for the equation of the regres- 
sion line of annual change in net earnings on annual change in 
gross earnings,