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 computed
 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 probability,
 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 follows:
 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 computed
 increase, + 1.08 — 0.497 AT, the residual is 4, and vice
versa. Hence, in Solution 7,, the observed decrease in net earnings
 was larger in Groups 1 and 3 than the increase computed
from (6). In Groups 2 and 4, the observed decrease in net earnings
 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 regression
 line of annual change in net earnings on annual change in
gross earnings,