A set of integers, S, contains more than one element. Is the range of S greater than its mean?
(1) S does not contain positive integers.
(2) The mean of S is negative.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(B) If S does not contain positive integers, as statement (1) defines, then S contains negative integers and/or zeroes.

If S consists of zeroes only, S = {0, 0, …, 0}, then its range is 0, because all the elements are the same. Besides, its mean is 0 as well, because the sum of the elements is 0. So in this case the mean and the range are equal.

If there is at least one non-zero element in S, then the sum will be negative (because S doesn't contain any positive numbers). The range is always a non-negative number, so it will be greater than the mean (which is negative).

Therefore statement (1) gives us two possible answers to the main question. Statement (1) by itself is NOT sufficient.

From Statement (2) we know that the mean is negative. Any range is a non-negative number. Therefore the range is greater than the mean. Statement (2) by itself is sufficient.

Since Statement (1) is insufficient and Statement (2) is sufficient, the correct answer is choice (B).

If this seems too abstract, it helps to pick some numbers, like the set {-5, -4, 2}, to see how the range will be greater than the mean. The mean is negative, and the range is 2 – (-5) = 7.
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I believe the answer should be E. Statement 2 should be insufficient, because what if the set is {-10,-15,-20}. The range would be -10 and the mean would be -15. In this case the mean is LARGER than its range..

Let us analyze the proposed set (-10,-15,-20) first.

It's range is -10 – (-20) = -10 + 20 = 10.
It's mean is ((-10) + (-15) + (-20)) / 3 = -45/3 = -15

10 > -15

The range is greater than the mean.

The range shows how far are the greatest and the least elements apart. Note, that range is always a NON-NEGATIVE number, i.e. positive or zero. And it equals zero only if all elements in a set are equal.

Statement (1) tells us that S contains no positive elements, but it still can contain 0, as 0 is an integer and it is neither positive, nor negative.
So S can be {0, 0, …, 0}.