Firstly, Sanju09.... thanks for posting interesting questions.

I am kind of stumped by this question, maybe because of lack of knowledge to differentiate the finer details of mean & median. The only rule I know is that when we are dealing with consecutive numbers & there are odd number of numbers then, Mean & Median are both same.

But, below is my reasoning on how I approach the question.

From the question, we know that

_ _ _ _ _ _ _ 130,000 _ _ _ _ _ _ _ = 15*150,000

_ _ _ _ _ _ _ 130,000 _ _ _ _ _ _ _ = 2,250,000

Statement I says: atleast one of the homes was sold for more than $165,000. Since the mean is $150,000 & median is $130,000, there should be homes greater than $150,000. So, this statement must be true.

Statement II says: $130,000 < atleast one home < $150,000. I am not sure how I can negate this statement. This statement may be true or false

Statement III says: at least one home < $130,000. Again, this statement may or may not be true, because all the costs below the median might be of the same price as the median.

Only Statement I provides a must be true condition.

Thanks Vemuri for cheering me, I don't know the meanings of this Indian name!

I understand your problem, so I am giving the explanation pretty early. See through it:

In order to solve the question easier, we simplify the numbers such as 150,000 to 15, 130,000 to 13, and so on.

I. Median is 13, so, the greatest possible value of sum of eight prices that are no more than median is 13*8 = 104. Therefore, the least value of sum of other seven homes that are greater than median is (15*15 - 104)/7= 17.3 > 16.5. It's true.

II. According to the analysis above, the price could be, 13, 13, 13, 13, 13,13,13,13, 17.3, 17.3, 17.3... So, II is false.

III. Also false.

Answer: only I must be true.