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Statistics - Kaplan- Approach

by imhimanshu » Tue Sep 13, 2011 4:34 am
Nine positive integers are contained in Set N. The median of Set N is the integer x, and all the values in Set N are 3x or smaller. What is the largest possible average(airthemetic mean) of Set N
1- x
2- 17x/9
3- (17x/9)-1
4- (17x-4)/9
5 - 3x

Please provide approach to the above question
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by sl750 » Tue Sep 13, 2011 4:42 am
We require the largest possible average.
The median of 9 positive integers is given as x
So the first four integers have to be x and the last four integers have to be as large as possible, in this case, 3x

x x x x x 3x 3x 3x 3x . Average is 17x/9
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by pavand » Tue Sep 13, 2011 5:03 am
Median = x
All values in N <= 3x
Largest possible average = Average of {x,x,x,x,x,3x,3x,3x,3x)
Average = (5x+12x)/9
=17x/9

Answer = 2
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by imhimanshu » Tue Sep 13, 2011 5:19 am
Thanks for posting solution.. but could you please detailed the below part -
Largest possible average = Average of {x,x,x,x,x,3x,3x,3x,3x)
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by sl750 » Tue Sep 13, 2011 5:56 am
HImanshu, the goal here to find the largest average possible. That is possible by ensuring that
the four integers above the median be as large as possible. In this case, we are told that it is 3x. Similarly, the integers below the median should be equal to the median in order to achieve the maximum average. If it were greater than the median the median would change

Or you could take numbers to help you understand better

say x=2

2 2 2 2 2 6 6 6 6 Avg = 34/9

if we let the first four integers to be any less than 2, you will see that the average will decrease
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by imhimanshu » Tue Sep 13, 2011 6:16 am
Hi sl750,
Thanks for replying. Appreciated
However,I am afraid to say again that I didn't get this..:-)
Is this a general rule or what?
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by GMATGuruNY » Tue Sep 13, 2011 8:48 am
imhimanshu wrote:Nine positive integers are contained in Set N. The median of Set N is the integer x, and all the values in Set N are 3x or smaller. What is the largest possible average(airthemetic mean) of Set N
1- x
2- 17x/9
3- (17x/9)-1
4- (17x-4)/9
5 - 3x

Please provide approach to the above question
To maximize the average, we need to maximize every value in Set N.
Let x=10.
Thus, the median -- or middle value -- is 10.

The 4 integers BELOW the median cannot be LARGER than the median of 10.
Thus, the largest possible combination of values for the 4 integers below the median is 10,10,10,10.

The 4 integers ABOVE the median can be any value greater than or equal to the median of 10.
The problem states that no value in the set is greater than 3x.
Since 3x = 3*10 = 30, no value in the set can be greater than 30.
Thus, the largest possible combination of values for the 4 integers above the median is 30,30,30,30.

Thus, the largest possible combination of values for set N = {10,10,10,10,10,30,30,30,30}.
Average = sum/number = (10+10+10+10+10+30+30+30+30)/9 = 170/9. This is our target.

Now we plug x=10 into the answers to see which yields our target of 170/9.

Only answer choice B works:
17x/9 = (17*10)/9 = 170/9.

The correct answer is B.
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by pavand » Wed Sep 14, 2011 2:21 am
imhimanshu,

In any number sequence, the way to find a median is:
Arrange the numbers in increasing order
The middle number in this increasing order is the median, if the number of items is odd
The average of the two middle numbers is the median, if the number of items is even

We have 3 pieces of information from the question
1. There are nine positive integers
2. Median of set N is x
3. All values in set N are 3x or smaller

From first two pieces & the question itself:
For the largest average of the sequence, all the numbers should be as large as possible
So, with 9 positive integers, there should be 4 integers before x for x to be the median. These 4 integers should be as large as possible. How do you build this?
They can be less than or equal to x itself
So N={x,x,x,x,x...}

Using the third piece of information:
N={x,x,x,x,x,3x,3x,3x,3x)
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by valto » Tue Apr 22, 2014 7:25 pm
Sorry to bring up this old thread again but I am not sure about one thing.
In the question it says "...and all the values in Set N are 3x or smaller" From this I am lead to believe that all other values in the set are either 3x or that they are smaller, not less than or equal to. When they say smaller I assume they can't equal x and therefore all other numbers before x are x-1 thus giving us an answer of (17x-4)/9.
I think the information given is a bit misleading.
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by [email protected] » Tue Apr 22, 2014 11:17 pm
Hi valto,

From what you typed, you acknowledge that the prompt states that "all values in Set N are 3X OR smaller"; this means that the values COULD be 3X or ANYTHING SMALLER than 3X. That factoid is the equivalent of "all values are less than, or equal to, 3X."

We're also told that the MEDIAN of Set N is X; this means that the "middle number" (the 5th number) in the set of 9 terms is X. This DOES NOT mean that the first 4 numbers must be less than X. It COULD mean that they're all less than X OR any number of them could EQUAL X and the rest are less than X.

Statistical terms have very specific definitions and you must know (and use) those definitions to answer the question correctly.

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Last edited by [email protected] on Wed Apr 23, 2014 12:21 pm, edited 1 time in total.
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by valto » Wed Apr 23, 2014 5:31 am
Hi Rich,

Thanks for the quick reply. It was my fault misreading the question. It was a long day of studying and my brain was tired :D
Anyways, the mistake I made was that in the sentence "...and all the values in Set N are 3x or smaller" I thought they meant that all values are either 3x or smaller than x, rather than smaller than 3x. Otherwise, the question is pretty straight forward.
Thanks again.
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