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QUEEN MARY, UNIVERSITY OF LONDON

MTH6116 Design of Experiments

Assignment 4 For discussion on 10th February 2011

Work on your solutions beforehand so you can participate in the discussion.

1 In Statistical Modelling II it was mentioned that the two-sample t-test (assum-ing equal variances in the populations) and the matched pairs or paired t-testare related to the analysis of variance for completely randomized designs andrandomized block designs respectively.

(a) Show that the test of H0 : 1 = . . . = t for equality of treatment effects

in a completely randomized experiment with t = 2 treatments and thetwo-sample t-test are equivalent.

(b) Similarly, show equivalence between the test of H0 : 1 = . . . = t fora randomized block design with t = 2 treatments and b blocks and thematched pairs t-test for b pairs.

Hint for the completely randomized design with t = 2 treatments: figure out howthe test statistic in the analysis of variance and the test statistic of the two-sample

t-test are related; also find out how the distributions of these two statistics are

related. Approach the part for the randomized block design in the same manner.

2 A group of ecologists is planning an experiment to compare 36 species of smallinsects for their effect in decomposing a mixture of leaves. Identical quantities ofthe leaf mixture will be put into 180 glass jars. For each species, several insectswill be put into each of five jars. The jars will then be covered with transparentfilm and left on the bench in the laboratory for several weeks before the amountof leaf decomposition is measured.

Because of the shape of the available bench space, the jars must be arranged ina 5 36 rectangle. One row of jars is next to the window and receives the mostsunlight; the fifth row receives the least. The ecologists suggest the followingdifferent methods of dealing with this difference in sunlight.

(a) Place the 180 jars in their positions in the rectangle in a haphazard manner.It is then likely that no species is especially favored.

(b) Start as above. Each morning, rearrange the jars within each column bymoving the jar nearest to the window to the position furthest from thewindow, and moving all the other jars in that column one place nearer tothe window. That way, differences in sunlight should be averaged for each

jar.

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(c) Treat each row as a block. Put one jar of each species in each row. In eachrow independently, choose a random permutation of 1, . . . , 36 and use it torandomize the positions of the jars in that row. The differences in sunlightwill be allowed for in the analysis.

(d) Put all five jars for each species into a single column, to ensure that eachspecies is exposed to the full range of sunlight conditions. Choose a singlerandom permutation of 1, . . . , 36 and use it to allocate species to columns.

Advise the ecologists of the advantages and disadvantages of these methods.

3 A physician wants to test a new drug to compare it with the current standarddrug. He would like to have 90% power for detecting a difference of 15 units ifhe does a two-sided hypothesis test at the 5% significance level. He believes thatin the population, 2 is about 100, for measurements in those units.

(a) If he uses r people for each drug, what is the smallest value of r that heshould use?

(b) What would be the smallest value ofr if there was a third drug (also beingreplicated r times)?

(c) With the design you proposed in (a), which would be the power to detecta shift of 10 units between treatments?

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