Dip DoE , A07
1
(a)
Solutions
1.1
The spread in all four samples appears normal, apart from possible exceptionally
high cases in samples A and C.
Membrane B appears best and Membrane C worst, with not much difference
between A and D.
(b)
(i)
The "Membrane" mean square measures the variation between the
sample means, plus chance variation.
(MS(Membrane) estimates 2 + Membrane effect)
The "Error" mean square measures chance variation within samples.
(MS(Error) estimates 2)
The F ratio measures the relative excess of between sample mean
variation over chance variation.
(F  1 + (Membrane effect)/2)
(ii)
(c)
(i)
Null hypothesis:
All four membrane types have the same mean
burst strength.
Critical value:
F.05, 3,36  2.85,
Conclusion:
15.54 > 2.85, null hypothesis is rejected.
Membrane B mean is significantly bigger than Membranes C and D
means and close to significantly bigger than Membrane A mean.
Membrane C mean is significantly smaller than the other three means.
Membranes A and D means are not significantly different.
(ii)
The simultaneous confidence intervals are slightly wider than confidence
interval for differences between individual pairs of means.
This is because the level of confidence in several intervals simultaneously
is reduced relative to the level of confidence in one of those intervals
individually. Widening the intervals increases the confidence. The extent
of widening may be chosen to compensate for the reduction in
confidence involved.
(d)
Membrane C can be eliminated from our inquiries. Membrane D shows no sign
of being an improvement on the existing Membrane A and so need not be
considered further. Membrane B shows some improvement on Membrane A but
not enough to recommend a change.
It may be worth while carrying out further comparisons between Membranes A
and B.
Dip DoE , A07
2.
(a)
Solutions
2.1
An interaction between two factors occurs when the effect on the response of
changing the level of one factor depends on the level of the other factor.
Summary:
Cooking in iron pots adds substantially to the average iron content of all cooked
foods. However, it adds considerably more to the iron content of meat, around
2.5 to 2.6 milligrams per 100gms on average, than to that of legumes or
vegetables, around 1.2 to 1.5.
The added iron content is very similar using aluminium and clay, (marginally
higher for clay), for all three food types.
Alternatively:
Vegetables tend to have lower iron content than either legumes or meat. Using
aluminium or clay pots, the differences are similar. Using iron pots the difference
from legumes is similar, while the difference for meat is much higher. The iron
content of meat is slightly lower than that of legumes using aluminium or clay
pots but considerably higher using iron pots.
(b)
Iron content includes an added contribution for each food type plus an added
contribution for each pot type plus an added contribution for each food type / pot
type combination, plus an added contribution due to chance variation.
Alternatively:
Iron content ( i , j, k) = overall mean + food type effect ( i ) + pot type effect ( j )
+ food/pot interaction effect ( i , j ) +  ( i , j, k )
i = Aluminium, Clay, Iron,
j = Meat, Legumes, Vegetables,
k = sample number 1,2,3,4.
or:
Yijk =  + i +  j + ij
ijk,
i = Aluminium, Clay, Iron,
j = Meat, Legumes, Vegetables,
k = sample number 1,2,3,4.
where Y is iron content,  represents food type effect,  represents pot type
effect,  represents the effect of interaction of food type and pot type and 
represents chance variation.
Summary:
The pot type and food type effects are very highly statistically significant and the
pot type / food type interaction is also highly statistically significant.
(c)
The adjustment makes no difference in this case because the experimental
layout is balanced, that is, the same number of samples is allocated to each pot
type / food type combination.
Dip DoE , A07
Solutions
2.2
Adjusted sums of squares are the additional sums of squares determined by
adding each particular term to the model given the other terms are already in the
model. When the terms in the model are uncorrelated, the presence or absence
of other terms has no effect on the sum of squares for a particular term.
However, when terms are correlated, the additional sum of squares determined
by adding each particular term to the model depends on which other terms are
already in the model. Terms will be correlated when the experimental layout is
unbalanced, in this case when there are unequal numbers of samples for
different pot type / food type combinations.
If the adjustment is not made, then the sums of squares associated with different
terms will depend on the order in which the terms are included in the model.
Thus, a term could appear to be statistically insignificant merely because another
term with which it was correlated was added first.
Dip DoE , A07
3.
(a)
Solutions
3.1
When there are possible differences between experimental units, units may be
grouped into blocks that are as similar to each other as possible, with estimates
of factor effects being made within each block and combined across blocks.
Randomisation involves randomly allocating treatments to units and to order of
experimentation with units.
The purpose of blocking is to eliminate the effects of known sources of
systematic variation between experimental units from the assessment of factor
effects.
The purpose of randomisation is to give protection against the presence of
unknown sources of systematic variation that may lead to differences between
experimental units. (Randomisation also provides a basis for valid statistical
inference).
In the experiment described, Figure 2 shows clear evidence of a decline in defect
rate over time. In the experiment carried out, days were blocked in pairs,
comparisons of new and old were made within each pair (block) and the
comparisons combined across pairs. Combined assessment via Figure 1 shows
that there was no significant difference between new and old. If blocking had not
been used and one process applied for the first four weeks and the other for the
second four, the observed difference that actually occurred, as shown in table 2,
would have been ascribed to a difference between processes.
In the experiment, allocation of new and old processes to days within pairs was
done systematically, with systematic switching of order to negate the effect of the
trend from one day to the next. An alternative approach would have been to
randomise the order within each day pair. This would have given protection
against the trend effect and also any other effect that might have been present
without the knowledge of the experimenters.
(b)
To illustrate the interaction effect, consider an experiment to assess the effects
of Training (Y,N) and use of a Script (Y,N) on the performance of telemarketing
sales personnel, as reflected in the percentage of sales calls that result in a sale.
The results are summarised below.
10.8
41.8
26.6
Yes
Script
–4.4
21.2
No
9.8
15.2
No
Training
20.6
Yes
Dip DoE , A07
(i)
Solutions
3.2
Suppose that 21 sales representatives were available. Using the
traditional "one-factor-at-a-time" approach, one trial will be run to
compare both levels of one factor, at the "standard" level of the second
and a second trial to compare the "standard" and "new" levels of the
second factor using the "best" level of the first factor. For example,
testing the script factor first, there will be two comparisons:
no script, no training,
script, no training,
followed by another,
best script, training.
It makes sense to allocate 7 sales representatives to each of the three
combinations involved, for balanced comparisons.
The statistical "multi-factor" approach will allocate sales representatives
to all four possible combinations. A balanced allocation would involve
assigning 5 sales representatives to each and either not allocating the
21st or allocating the 21st arbitrarily to one of the combinations.
Either way, a comparison of Script with No Script now involves a
comparison of at least 10 results with at least 10, rather then 7 with 7 as
in the "one-factor-at-a-time" approach and similarly with the comparison
of Training with No Training. For the purpose of identifying the best
levels of the two factors separately, this is a more efficient use of
resources.
(ii)
Note that, if the traditional approach sequence in part (i) is followed, the
first comparison will lead to using No Script as the best level of the first
factor and the second comparison will lead to using No Script and
Training as the best combination. Clearly, the best combination of Script
and Training will be found by the statistically recommended "multi-factor"
approach. It would not have been found by the "one-factor-at-a-time"
approach as outlined in part (i) although it would if we happened to test
the Training factor first. Having to depend on the choice of factor to test
first in order to discover the best combination of factor levels can hardly
be described as good science.
N.B. Other illustrations may be used.