richs_ca wrote:What is the average heigh of the n people in a certain group?
1) The average heigh of the n/3 tallest people in the group is 6 feet 2.5 inches, and the average height of the rest of the people in the group is 5 feet 10 inches.
2) The sum of the heights of n people is 178 feet 9 inches.
I'm a bit confused, because 1) doesn't seem to give the total number of people in the group, yet that's the answer.
statement (1) is a
weighted average: it's just like a normal average, except in that you must account for the # of times each data point appears. so here's how you set it up:
average = (sum of all heights) divided by (number of people)
= ( (n/3)(74.5 inches) + (2n/3)(70 inches) ) divided by (n)
no need to do the arithmetic here; just look at it,** and notice that you can pull a common factor of 'n' out of everything on top, after which you can strike the 'n's and leave a purely numerical expression.
**you might need to write it out on a piece of paper to see this, because fractions and forums just don't mix well.
--
there's also a cool shortcut for weighted averages: if a certain
fraction of a population has statistic x and the rest has statistic y, then the average is the
same fraction of the way from y to x.
so, for instance, in this problem, 1/3 of the population is 74.5 inches tall, and the remaining 2/3 of the population is 70 inches tall. so, the average is 1/3 of the way from 70 to 74.5 (or, if you want, 2/3 of the way from 74.5 to 70).
one consequence of the foregoing fact is that
any weighted average involving
definite fractions of a population will work out to the same number, regardless of the number of individuals in the population. this is a truly valuable observation for data sufficiency.
--
you probably already figured this out, but (2) is insufficient because you need to know how many people there are.
so answer = a
