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Suppose that one has data from a randomized-block experimental design,
for which 5 treatments are compared using 8 blocks. If Friedman's test
is used to test the null hypothesis of no differences between treatments
against the general alternative, what is the expected value of the
average rank for each treatment if the null hypothesis is true?
Within each block, the treatments are ranked from 1 to 5.
If the null hypothesis is true, each of the possibilities, 1, 2, 3, 4,
and 5, are equally likely to be assigned to each of the treatments,
which gives us that the expected value of the rank assigned to a particular
treatment is 1(1/5) + 2(1/5) + 3(1/5) + 4(1/5) + 5(1/5) = (1 + 2 + 3 + 4 + 5)/5 = 3 (just the average of the values, 1, 2, 3, 4, and 5), which gives us that the expected value of the average rank for each
treatment is also 3.
- [ A ] 2.5
- [ B ] 3
- [ C ] 4
- [ D ] 4.5
- [ E ] 5
- [ F ] 27
- [ G ] 27.5
- [ H ] 28
- [ I ] 35
- [ J ] 40