by request!
i was asked to explain the following further:
lunarpower wrote:the answer is (c) if you have the following modified version of the problem:
what is the median of a set of numbers?
(1) more than half of the numbers are > 1000.
(2) more than half of the numbers are < 1000.
THIS problem has answer (c).
here's the reason why.
in the following, imagine that you put the numbers in order from left to right.
for clarity, i'll split the discussion into two cases: (1) sets with an ODD number of data, and (2) sets with an EVEN number of data.
-- CASE 1: sets with an odd number of data --
in this case, there is one middle value (median); let's call it M.
statement 1:
say the orange numbers are
> 1000:
# # # ... # # M
# # ... # # #
in this case, LESS than half of the numbers are
> 1000.
now, say the orange numbers are
> 1000:
# # # ... # #
M # # ... # # #
in this case, MORE than half of the numbers are
> 1000.
taking these two examples together, you should be able to see that,
if more than half of the data are > 1000, then the median must be > 1000.
therefore,
statement 1 implies that the median > 1000.
this is still insufficient to answer the problem, though, because the median could be any number
> 1000.
statement 2:
say the orange numbers are
< 1000:
# # # ... # # M # # ... # # #
in this case, LESS than half of the numbers are
< 1000.
now, say the orange numbers are
< 1000:
# # # ... # # M # # ... # # #
in this case, MORE than half of the numbers are
< 1000.
taking these two examples together, you should be able to see that,
if more than half of the data are < 1000, then the median must be < 1000. note that this argument is exactly symmetric to the argument used to interpret statement 1.
therefore,
statement 2 implies that the median < 1000.
this is still insufficient to answer the problem, though, because the median could be any number
< 1000.
if we have both statements together, then the median is both
> 1000 and
< 1000. the only way that can be true is if the median is actually 1000.
therefore (c).
