# Last month 15 homes were sold in town x. the average (arithmetic mean) sale price of the homes was \$150,000 and the median sale price

Last month 15 homes were sold in town x. the average (arithmetic mean) sale price of the homes was \$150,000 and the median sale price was \$130,000. which of the following statements must be true? i. at least one of the homes was sold for more than \$165,000.ii. at least one of the homes was sold for more than \$130, and less than \$150,000iii. at least one of the homes was sold for less than \$130,000.a. i onlyb. ii onlyc. iii onlyd. i and iie. i and iii

A. 1 only. Step-by-step explanation:

A. I Only. Step-by-step explanation: To begin, we must first be clear that it is the median and that it is the arithmetic mean:
Median is the middle value of a sequence of ordered numbers, for example:
{4,4,4,4,4}, the median is 4 despite being the same numbers.
Now the arithmetic mean is the average value of the samples and is independent of the amplitudes of the intervals.
Then let’s analyze each of our options:
I. At least one of the homes was sold for more than \$ 165,000.
We know through the flushed:
X1 + X2 +. . . + X7 + (X8 = \$130,000) + X9 +. . . + X15 = 15 ∗ 150,000 = \$ 2,250,000
Now we will assume the lowest possible value from X1 to X8 = \$ 130,000 and from X9 to X15 = X, which is what we want to calculate. That is to say:
X1 = X2 = X3 = X4 = X5 = X6 = X7 = X8 = 130 and X9 = X10 = X11 = X12 = X13 = X14 = X15 = X, knowing that the total value must be the average of 15, which is equal to \$ 2250000 , we have the following equation:
8 ∗ \$ 130,000 + 7X = \$ 2,250,000
Rearranging:
X = (\$ 2,250,000 \$ – \$ 1,040,000) / 7
X = \$ 172,857
Therefore the first statement is true, because at least one house was sold at \$ 172,857 which is more than \$ 165,000
Evaluating the second option
II. At least one of the homes was sold for more than \$ 130,0000 and less than \$ 150,000
As the example of the median in the previous case you could have 8 houses that were sold for \$ 130,000 or less, therefore here it loses validity, statement II is false.
Evaluating the third option
III. At least one of the homes was sold for less than \$ 130,000.
We know that the eighth house sold for \$ 130,000, but houses 1 to 7 may also have been sold for that same price. The statement III is false.
Therefore the answer is A. I Only.

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