Data Sufficiency

abhirup1711 wrote:If set S consists of even number of integers, is the median of set S negative?
1.Exactly half of all elements of set S are positive.
2.The largest negative element of set S is -1.

NOTE: Since set S contains an even number of elements, the median will be the average of the two middlemost values.

Target question: Is the median of set S negative?

Statement 1: Exactly half of all elements of set S are positive.
There are several sets that meet this condition. Here are two:
Case a: set S = {-5, 1}, in which case the median is negative
Case b: set S = {-1, 1}, in which case the median is not negative
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The largest negative element of set S is -1.
There are several sets that meet this condition. Here are two:
Case a: set S = {-3, -1}, in which case the median is negative
Case b: set S = {-1, 6}, in which case the median is not negative
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
There are two cases to consider: set S contains at least 1 zero, and S contains no zeros.

Set S contains at least 1 zero: Since half of the elements are positive, we can draw a line in the middle of set S to get: {some values less than or equal to zero, 0, | positive integers}. Here, the median will be the average of 0 and the smallest positive integer. As such, the median will be positive. In other words, the median cannot be negative.

Set S contains no zeros: Since half of the elements are positive, we can draw a line in the middle of set S to get: {some negative values, -1, | positive integers}. Here, the median will be the average of -1 and the smallest positive integer. Since the smallest positive integer must be greater than or equal to 1, the median will be greater than or equal to zero. In other words, the median cannot be negative.

Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

Fighton!! wrote:I think first case(with "at least one zero") in above post is never possible . Because 0 is neither +ve nor -ve. if you add 0 in the set then it will never form a set of even no of integer,given that exactly half of integers are positive. if exactly half of the integers are positive, say 2x in numbers and other half is negative, say 2y in numbers, and then u add zero in the set. it will form the set of odd number of integers with total 2x+2y + 1(even + odd + zero) integers, contradicting the scenario that S is the set of even no of integers.

The part above, in green, is not true.
For example, in the set {-1, 0, 1, 1}, 1/2 of the elements are positive, yet only 1/4 of the elements are negative.

Cheers,
Brent