Official Solution:
First of all:
The median of a set with odd # of terms is just a middle term (when ordered in ascending/descending order).
The median of a set with even # of terms is the average of two middle terms (when ordered in ascending/descending order).
Next, one of the most important properties of evenly spaced set (aka arithmetic progression):
In any evenly spaced set the arithmetic mean (average) is equal to the median and can be calculated by the formula \(mean=median=\frac{a_1+a_n}{2}\), where \(a_1\) is the first term and \(a_n\) is the last term. Given the set \(\{7,11,15,19\}\), \(mean=median=\frac{7+19}{2}=13\).
(1) All members of \(S\) are consecutive multiples of 3. This statement says that \(S\) is an evenly spaced set, thus its mean equals to its median. Sufficient.
(2) The sum of all members of \(S\) equals 75. Clearly insufficient, consider two sets \(\{25, 25, 25\}\) and \(\{0, 0, 75\}\).
Answer: A