Solution file for Additional exercise 7.5
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Part I:
No additional solution given because the text is a sort of solution
itself. It is certain though that I would never have come up with the
refinements of the experimental procedure described. :-)

Part II:
(a) The objectives of an experiment on detergents is not entirely clear.
One could do a comparative study assessing several brands or concentrate
on a single product and the way performance depends on the factors
listed. For this solution, the latter is taken to be the intended
objective.

(b) The text lists 5 factors that should be considered as sources of
variation. It is hardly feasible to study them all simultaneously so
some restrictions on the scope of the experiment are necessary. The
concentration and the temperature seem the most important ones, and also
the ones that can most easily be varied, so those will be the treatment
factors. For simplicity, only two levels of each factor (say C1/C2 and
T1/T2) will be considered, giving four different treatment combinations:
C1T1, C1T2, C2T1, C2T2. The hardness of the water needs to be fixed unless
one works with different water supplies - which seems rather complicated.
The experiment can be repeated in other cafeterias with different water
hardness, if desired. The type of soil is controlled by blocking on the
dish type, as described in the text. One might want to standardize the
dish washing by using only a single operator. This may cause a substantial
time delay before the dishes are washed for some treatments, but if there
is a continuous flow of plates available that might not be a problem.
If several operators are involved it would be desirable to block also on
operators.

The experimental unit is one wash basin full of water (of a given temperature)
and soap (of a given concentration), with the plates to be washed in the basin.
The random variation between experimental units stems primarily from the
variation in the degree of soiling of the plates. Perhaps the energy
level of the operator factors in as well.

The two main blocking factors, soil type and operator, were already
discussed above. There are no obvious noise factors (factors that are
measured and included in the model instead of blocked upon). None of the
blocking factors seem to allow easy quantification into a noise factor
that could be included in the model. Also, there are no supplementary
measurements for each experimental unit available (covariates).

(c) The actual experimental design depends on whether blocking on
operators is necessary or of interest. If no, the four treatments are
randomly assigned to the experimental units for each course, thereby
determing the order in which the four treatments will be measured. This
is a completely randomized block design.

If yes, the operators should form a second blocking. One easy way to
achieve that is to use data for 4 courses (or 8, 12, etc), and to assign
treatments to operators within the courses by a Latin square. The rows
correspond to course number, the columns to operator, and the symbols to
the four treatments. As a minimum, randomization should involve randomly
permuting operators and treatments (it is not necessary to permute all
three factors).

(d) The measurements to be made are the times until the foam is reduced
to a thin surface layer. The experiment will probably need a referee
type person to decide when the soap of a wash basin is sufficiently thin
to stop. The referee person should be blinded to the actual treatments.
It might be possible to let the operators make that judgement themselves,
thereby leading to increased operator variability, but there are
concerns with regards to bias (the operators can hardly be blinded to
the treatments) and validity (if the operators interpret the outcome
very differently, what do the measurements really mean).