Catch: 0 can be present in the set.

Statement (1) says that half of the numbers are positive and the rest half of the numbers are =< 0. The median will the mean of the largest negative number (or zero) and the smallest positive number. But, we don't know their values.
INSUFFICIENT.

Statement (2) just says that the largest negative value is - 1. Very little data here.
INSUFFICIENT.

Lets try to combine them.
The largest negative value is - 1. Half of the integers are positive.
If 0 is present in the set then median = (-1 + 0) / 2 = -0.5. NEGATIVE
If the smallest postive value is 1, median = 0. NEUTRAL
if the smallest positive value is more than 1, median > 0. POSITIVE

Is the median negative? May be. May be not.
No conclusive answer.
(E) should be the correct option

Aneesh,

I reached the same anwer. However as per the explanation given - since 0 is neither + or -, it is for sure not -ve.

This is what explanation says after (1)+(2)

(1) + (2) half positive, biggest negative -1 --> median is either (0 + positive)/2=positive (so not negative) or (-1 + at least smallest positive integer, which is 1)/2=0 (min value of median in this case) also not negative.

Yes. makes sense.
In the case when 0 is present, the median is between 0 and the positive number because these are the two middle values now. Hence, the median is positive.

(C) should be correct.

Good problem!