kbm1975 is right, but I would like to clarify the answer a little bit more.
When I approach questions like this, I focus on the phrase "which must be true?" Thinking like a lawyer, I ask myself if I can come up with a counter-example in which it is not true.
First, I try to come up with a counter example for I. In essence, I want to minimize the price of the most expensive home. In order to do that, I want the homes to be in as tight of a price range as possible. When a home sells for less than the mean, then other homes are going to have to make up the difference. Ok, so let me look at my requirements. The median price is 130K. Since there are 15 homes, the 8th most expensive home had to have sold for 130K. Because I want to keep the price of the homes in a tight band around the mean, I will say that the 8 least expensive homes sold for 130K.
Now, we have the 7 most expensive homes to price. Again, keeping with my strategy of keeping the price as close to the mean as possible, I sell these houses all for the same price. To solve for that price, I have the equation:
((8 x 130) + 7x)/15 = 150
I solve for X and get $172, 857. What this means is that the lowest I can price the most expensive house (and still have mean 150K, median 130K). In other words, at least one house has to sell for this amount or more... so there is no counter example to I and I holds.
Moving on to II, you can see that we already came up with a scenario where 8 houses were 130K and 7 houses were 172K, so II does not necessarily hold. (Yes, you could make a case where it does hold... but remember, one counter example destroys the requirement that it "must be true.")
Moving onto III, you can see our scenario also provides a counter example, so it is not necessarily true.
Thus, only I is true and the answer is A.
Feel free to ask for clarification, as always.
Tatiana
Tatiana Becker | GMAT Instructor | Veritas Prep
