; s 
the averages of the two samples are therefore 2 and 5 respectively; and the 
variance of their difference is (‘2 + 5) s* = "05>. We may now proceed to 
test the hypothesis that all seven pieces could in fact be derived from one film 
(also of variance s°), ie. that the difference between the two samples is no more 
than can be expected to arise from random sampling from one and the same 
film. This hypothesis can be tested very readily by Student's t-test, by calculating 
the quantity £, — (x — YW) 5, where x and x' are the mean distortions of the 
two samples. Since the means are calculated from the data, the number of degrees 
of freedom associated with the quantity #, is(2— 1) + (5— 1) = 5. The t-tables 
then give the probability that differences equal to or greater than X — X^ could 
arise from random sampling. Suppose now that we did not know the actual sizes 
of the samples, but only that they ranged from two to five. We may assume that 
the samples were equal and consisted of say four pieces each, which would give 
£, — (x — x)! + 4) = (x —x)/(tsy^s, with (4—1)--(4—1) = 6 
degrees of freedom. If t, — 2.572, the difference would be significant on the 5 per 
cent level, because such a value of t can arise by chance only in 5 out of one 
hundred such sample pairs. The correct probability level can of course be worked 
out by using the correct sample sizes, giving t, — (!/o X !9/;) t, — 2.063 with its 
five degrees of freedom. This is not significant on the 5 per cent level, but it passes 
the 10 percent point. 
Whether or not we decide to drop further investigations in view of this 
evidence of insignificance depends entirely on the envisaged consequences. The 
point is that we have lost the chance of knowing the exact probability because the 
actual sizes of the samples are not given *). 
(b). The average shrinkage is given without the respective standard error. A 
general statement is however made that “absolute errors in tests made at different 
times do not normally exceed +.03 per cent and the relative errors between the 
two sets of samples tested together are approximately +.01 per cent of the 
dimension". I have underlined the words ‘normally’ and ‘approximately’ because 
they deprive the given numbers of any clear significance. The words imply that 
the numbers are not the observed ranges of error, and as we do not know the 
proportions of the examined cases to which they refer, we are not in a position to 
make good use of them. 
Incidentally, it is very important to give the standard error to a sufficient 
number of decimal places to avoid unnecessary errors in the probability levels. A 
standard error rounded off to .01 may be anything between say .014 and .006. 
The error of rounding off can therefore be two thirds the correct value, or even 
greater. The value of # can be as much as 1°/s or as little as /7 its ‘approximate’ 
value. Thus, the approximate value of the probability level would be given as 5 
per cent whether the true value was 20 per cent or 1 per cent. 
2:2. The second remark is more related to the intrinsic value of the work. A 
few examples will emphasize the care which must be exercised to ensure that 
nothing which may influence the results of the experiment is overlooked. 
*) The analysis of samples of unequal sizes is sometimes difficult so that adherence to equal sizes 
is a practical advantage. 
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