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Was the average (arithmetic mean) sale price of a new home in region R

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Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   09 Jun 2023, 11:33
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Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000 ?

(1) Last month the median sale price of a new home in region R was at least $100,000.

(2) Last month the sale prices of new homes in region R ranged from $75,000 to $150,000.
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Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   10 Jun 2023, 12:10
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carcass wrote:
Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000 ?

(1) Last month the median sale price of a new home in region R was at least $100,000.

(2) Last month the sale prices of new homes in region R ranged from $75,000 to $150,000.


Statement 1

(1) Last month the median sale price of a new home in region R was at least $100,000.

The information with reference to the median price of the house will not help us find the average as we don't have information on the prices of the other houses.

For example -

3 houses were sold last month. Set A represents the selling price of the houses

Case 1

A = {1, 100000, 100001} → The average, in this case, is less than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - No

Case 2

A = {100000, 100000, 100000000000} → The average, in this case, is greater than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - Yes

The statement alone is not sufficient, we can eliminate A and D.

Statement 2

(2) Last month the sale prices of new homes in region R ranged from $75,000 to $150,000.

Similar to statement 1, statement 2 is also not sufficient as all the values of the houses sold last month can lie either on the lower side of the range or on the upper side of the range. Therefore we can have multiple answers to the target question.

Case 1

A = {75000, 75000, 75000} → The average, in this case, is less than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - No

Case 2

A = {150000, 150000, 150000} → The average, in this case, is greater than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - Yes

The statement alone is not sufficient, we can eliminate B.

Combined

The statements combined don't help as the average can be a value between $75,000 and $150,000.

Case 1

A = {75000, 100000, 100000} → The average, in this case, is less than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - No

Case 2

A = {100000, 100000, 150000} → The average, in this case, is greater than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - Yes

Option E
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Re: Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   10 Jun 2023, 23:35
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gmatophobia wrote:
carcass wrote:
Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000 ?

(1) Last month the median sale price of a new home in region R was at least $100,000.

(2) Last month the sale prices of new homes in region R ranged from $75,000 to $150,000.


Statement 1

(1) Last month the median sale price of a new home in region R was at least $100,000.

The information with reference to the median price of the house will not help us find the average as we don't have information on the prices of the other houses.

For example -

3 houses were sold last month. Set A represents the selling price of the houses

Case 1

A = {1, 100000, 1100001} → The average, in this case, is less than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - No

Case 2

A = {100000, 100000, 100000000000} → The average, in this case, is greater than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - Yes

The statement alone is not sufficient, we can eliminate A and D.

Statement 2

(2) Last month the sale prices of new homes in region R ranged from $75,000 to $150,000.

Similar to statement 1, statement 2 is also not sufficient as all the values of the houses sold last month can lie either on the lower side of the range or on the upper side of the range. Therefore we can have multiple answers to the target question.

Case 1

A = {75000, 75000, 75000} → The average, in this case, is less than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - No

Case 2

A = {150000, 150000, 150000} → The average, in this case, is greater than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - Yes

The statement alone is not sufficient, we can eliminate B.

Combined

The statements combined don't help as the average can be a value between $75,000 and $150,000.

Case 1

A = {75000, 100000, 100000} → The average, in this case, is less than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - No

Case 2

A = {100000, 100000, 150000} → The average, in this case, is greater than 100000.

Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000? - Yes

Option E


Typo: A = {100000, 100000, 150000}
--> A = {75000, 100000, 150000}

Typo:
A = {100000, 100000, 150000}
-->
A = {75000, 100000, 150000}
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Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   10 Jun 2023, 23:58
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zhanbo wrote:

Typo: A = {100000, 100000, 150000}
--> A = {75000, 100000, 150000}

Typo:
A = {100000, 100000, 150000}
-->
A = {75000, 100000, 150000}


zhanbo - Are you referring to Combined - Case 2?

I don't suppose there is a typo in the example provided.

We know that the sale prices of new homes in Region R ranged from $75,000 to $150,000. So the price of each house sold was between $75,000 and $150,000.

Assume two houses were sold for $1M, and one house was sold for $1.5M → the average price sale price of a new home = $1.16M.

We don't necessarily need to include $75,000 in the set to prove that the average price of the house was at least $1M. IMO, we can use any value within the mentioned range to prove that the statements combined are not sufficient. Let me know if I am missing something :)
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Re: Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   11 Jun 2023, 00:20
gmatophobia wrote:
zhanbo wrote:

Typo: A = {100000, 100000, 150000}
--> A = {75000, 100000, 150000}

Typo:
A = {100000, 100000, 150000}
-->
A = {75000, 100000, 150000}


zhanbo - Are you referring to Combined - Case 2?

I don't suppose there is a typo in the example provided.

We know that the sale prices of new homes in Region R ranged from $75,000 to $150,000. So the price of each house sold was between $75,000 and $150,000.

Assume two houses were sold for $1M, and one house was sold for $1.5M → the average price sale price of a new home = $1.16M.

We don't necessarily need to include $75,000 in the set to prove that the average price of the house was at least $1M. IMO, we can use any value within the mentioned range to prove that the statements combined are not sufficient. Let me know if I am missing something :)


My interpretation of the statement "the sale prices of new homes in region R ranged from $75,000 to $150,000" means
that the minimum sale price for new homes in region R is $75,000 and
that the maximum sale price for new homes in region R is $150,000.

My interpretation is influenced by the definition of range = maximum - minimum.

Your inference that 75000 and 150000 are only possible (but not necessarily actual) upper and lower threshold may be valid. I would like to hear opinions from math experts. carcass
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Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   11 Jun 2023, 00:29
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zhanbo wrote:
My interpretation of the statement "the sale prices of new homes in region R ranged from $75,000 to $150,000" means
that the minimum sale price for new homes in region R is $75,000 and
that the maximum sale price for new homes in region R is $150,000.

My interpretation is influenced by the definition of range = maximum - minimum.

Your inference that 75000 and 150000 are only possible (but not necessarily actual) upper and lower threshold may be valid. I would like to hear opinions from math experts. carcass


I would like to correct (or possibly clarify) the highlighted interpretation. I am not assuming that the $75,000 and $150,000 are the ONLY values. If that were the case, I wouldn't have used $100,000 in my example.

My interpretation ⇒ The price of each of the houses sold was between $75,000 and $150,000. The price can take any value between the upper limit and the lower limit.

Ex: In my country, the average temperature in June ranges from 30 to 40 degrees Celsius. → The average temperature can be any value between 30 degrees to 40 degrees. It 'may' (not necessarily though) be 30 degrees at the lowest and 40 degrees at the highest. However, it can also be 35 degrees, or 35.5 degrees and so on.
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Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   11 Jun 2023, 02:50
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Thanks to mention me as a math expert sir. zhanbo

Unfortunately, I am not, Bunuel is. On the verbal side probably........

Nonetheless

Following is the official explanation

Given that the median price was at least $100,000, the following two examples show that it cannot be determined whether the average price was at least $100,000.
Example 1: Average price is greater than $100,000



$75,000-----$100,000---------$150,000


$100,000--------------------------------$150,000


$100,000---------------------------------$150,000



The median of these 7 prices is $100,000 and the average of these prices is greater than $100,000, since the sum of these 7 prices is 7($100,000) + (−$25,000 + $50,000 + $50,000 + $50,000), which is greater than 7($100,000).
Example 2: Average price is less than $100,000



$75,000 -- $100,000 --- $150,000


$100,000 -----------------------------$150,000


$100,000-------------------------------$150,000



The median of these 7 prices is $100,000 and the average of these prices is less than $100,000, since the sum of these 7 prices is 7($100,000) + (−$25,000 − $25,000 − $25,000 + $50,000), which is less than 7($100,000);

NOT sufficient.
Given that the prices ranged from $75,000 to $150,000, the same examples above show that it cannot be determined whether the average price was at least $100,000; NOT sufficient.

Taking (1) and (2) together, it cannot be determined whether the average price was at least $100,000 because the two examples above each satisfy both (1) and (2).
The correct answer is E;
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Re: Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   11 Jun 2023, 03:57
carcass wrote:
Thanks to mention me as a math expert sir. zhanbo

Unfortunately, I am not, Bunuel is. On the verbal side probably........

Nonetheless

Following is the official explanation

Given that the median price was at least $100,000, the following two examples show that it cannot be determined whether the average price was at least $100,000.
Example 1: Average price is greater than $100,000



$75,000-----$100,000---------$150,000


$100,000--------------------------------$150,000


$100,000---------------------------------$150,000



The median of these 7 prices is $100,000 and the average of these prices is greater than $100,000, since the sum of these 7 prices is 7($100,000) + (−$25,000 + $50,000 + $50,000 + $50,000), which is greater than 7($100,000).
Example 2: Average price is less than $100,000



$75,000 -- $100,000 --- $150,000


$100,000 -----------------------------$150,000


$100,000-------------------------------$150,000



The median of these 7 prices is $100,000 and the average of these prices is less than $100,000, since the sum of these 7 prices is 7($100,000) + (−$25,000 − $25,000 − $25,000 + $50,000), which is less than 7($100,000);

NOT sufficient.
Given that the prices ranged from $75,000 to $150,000, the same examples above show that it cannot be determined whether the average price was at least $100,000; NOT sufficient.

Taking (1) and (2) together, it cannot be determined whether the average price was at least $100,000 because the two examples above each satisfy both (1) and (2).
The correct answer is E;


carcass, thanks for the OG explanation, which appears to interpret "range from 75,000 to 150,000" strictly, meaning
that the minimum sale price for new homes in region R is $75,000
and that the maximum sale price for new homes in region R is $150,000.
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Re: Was the average (arithmetic mean) sale price of a new home in region R [#permalink] New post   19 Jan 2024, 11:07
Quote:
Was the average (arithmetic mean) sale price of a new home in region R last month at least $100,000 ?

(1) Last month the median sale price of a new home in region R was at least $100,000.

(2) Last month the sale prices of new homes in region R ranged from $75,000 to $150,000.


Let's make it polarized.
Red indicates the median.

{75, 100 ... 100, 100, 150 ... 150}
The average of this set is definitely more than 100.

{75, 75 ... 75, 100, 100 ... 100, 150}
As long as there're adequate 75 and 100 before 150, the average of this set is converging to a number more than 75 but less than 100.
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Re: Was the average (arithmetic mean) sale price of a new home in region R [#permalink]
19 Jan 2024, 11:07
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