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The random variable X is normally distributed

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The random variable X is normally distributed [#permalink] New post 24 Dec 2015, 07:44
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The random variable X is normally distributed. The values 650 and 850 are at the 60th and 90th percentiles of the distribution of X, respectively.

Quantity A
Quantity B
The value at the 75th percentile of the distribution of X
750


A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.


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Difficulty: hard
[Reveal] Spoiler: OA

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Re: The random variable X is normally distributed [#permalink] New post 24 Dec 2015, 07:53
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Now this is a question that essentially tests our understanding of the bell curve.

Image

Now given above is a standard bell curve. When we say value of 60%tile is 650 we mean that it is the area under the bell curve from 0%tile to 60%tile. And
When we say value of 90%tile is 850 we mean that it is the area under the bell curve from 0%tile to 90%tile. Now 75%tile is exactly in the middle between 60%tile and the 90%tile. However there is more area of the bell curve under 60%tile to 75%tile than 75%tile to 90%tile.

Hence the value of 75th %tile should be lower to than 750 (which is half way between 650 and 850).


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Re: The random variable X is normally distributed [#permalink] New post 14 Jun 2016, 11:10
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Isn't that the other way around?
The value corresponding to the 75th percentile will be closer to the mean and 650 and further from 850. Thus this value will be less that 750 => B
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Re: The random variable X is normally distributed [#permalink] New post 14 Jun 2016, 13:38
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thanosro wrote:
Isn't that the other way around?
The value corresponding to the 75th percentile will be closer to the mean and 650 and further from 850. Thus this value will be less that 750 => B



Yes you are correct! B is greater.


Thanks for the correction.
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Re: The random variable X is normally distributed [#permalink] New post 02 May 2017, 11:54
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sandy wrote:
The random variable X is normally distributed. The values 650 and 850 are at the 60th and 90th percentiles of the distribution of X, respectively.

Quantity A
Quantity B
The value at the 75th percentile of the distribution of X
750




VERY tricky question.

When it comes to normal distributions, it might be useful to think of the area under the curve as the ENTIRE population being examined.
Image
In fact, it helps more of you think of each blue pixel/dot under the curve as representing one member of the population.


So, if a score of 650 is at the 60th percentile, we know that 60% of all of the blue pixels/dots lie to the left of 650
Image


Likewise, if a score of 850 is at the 90th percentile, we know that 90% of all of the blue pixels/dots lie to the left of 850
Image



IMPORTANT: If we draw a new line that takes the population BETWEEN the 60th and 90th percentiles and divides that population into two EQUAL populations, then that new line will represent the 75th percentile score.
Image
So, where should that line go?

Well, if we draw it halfway between 650 and 850...
Image
...then more of the population will be on side A.
So, this line will NOT divide the population between the 60th and 90th percentiles into two equal populations.


To get a line the DOES divide the population into two equal populations, we need to draw it right about here...
Image


Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850)
Image

So, the SCORE associated with the 75th percentile is LESS THAN 750

Answer: B
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Re: The random variable X is normally distributed [#permalink] New post 30 Jul 2017, 13:59
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Nice explanation with graphics. Clear and clever.! Thanks.
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Re: The random variable X is normally distributed [#permalink] New post 06 Sep 2018, 15:20
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I still do not understand how I can solve this type of a question on a graph. I mean visuals are great for practicing and understanding at our own time but in exam, we need to use solid concept and logic for a like aminute or so. I feel like hand drawn thing is a risky guess on exam. So does someone have a better strategy for exam please?
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Re: The random variable X is normally distributed [#permalink] New post 07 Sep 2018, 15:37
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kruttikaaggarwal wrote:
I still do not understand how I can solve this type of a question on a graph. I mean visuals are great for practicing and understanding at our own time but in exam, we need to use solid concept and logic for a like aminute or so. I feel like hand drawn thing is a risky guess on exam. So does someone have a better strategy for exam please?


I would argue that Brent is spot on with the best method to solve this problem.

The graph in this case is the logic. In order to uunderstant percentiles and probability distribution you need to understnad what percentile mean on the graph and what a probabaility distribution looks like.

Unfortunately, there is no short cut to solve these problems and these tend to be one of the most difficult problems on the GRE.
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Re: The random variable X is normally distributed [#permalink] New post 10 Sep 2018, 09:44
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Very nice explanation
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Re: The random variable X is normally distributed [#permalink] New post 14 Jan 2019, 01:10
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GreenlightTestPrep wrote:
sandy wrote:
The random variable X is normally distributed. The values 650 and 850 are at the 60th and 90th percentiles of the distribution of X, respectively.

Quantity A
Quantity B
The value at the 75th percentile of the distribution of X
750




VERY tricky question.

When it comes to normal distributions, it might be useful to think of the area under the curve as the ENTIRE population being examined.
Image
In fact, it helps more of you think of each blue pixel/dot under the curve as representing one member of the population.



this explanation was super helpful.
The halfway point between scores is flat, and can be equal, but if you drew a line from the 750 score to its percentile, it would have to be less than halfway between 650 and 850, since the curve curves downward, with less area.

So, if a score of 650 is at the 60th percentile, we know that 60% of all of the blue pixels/dots lie to the left of 650
Image


Likewise, if a score of 850 is at the 90th percentile, we know that 90% of all of the blue pixels/dots lie to the left of 850
Image



IMPORTANT: If we draw a new line that takes the population BETWEEN the 60th and 90th percentiles and divides that population into two EQUAL populations, then that new line will represent the 75th percentile score.
Image
So, where should that line go?

Well, if we draw it halfway between 650 and 850...
Image
...then more of the population will be on side A.
So, this line will NOT divide the population between the 60th and 90th percentiles into two equal populations.


To get a line the DOES divide the population into two equal populations, we need to draw it right about here...
Image


Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850)
Image

So, the SCORE associated with the 75th percentile is LESS THAN 750

Answer: B
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Re: The random variable X is normally distributed [#permalink] New post 02 Jun 2019, 11:45
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GreenlightTestPrep wrote:
sandy wrote:
The random variable X is normally distributed. The values 650 and 850 are at the 60th and 90th percentiles of the distribution of X, respectively.

Quantity A
Quantity B
The value at the 75th percentile of the distribution of X
750




VERY tricky question.

When it comes to normal distributions, it might be useful to think of the area under the curve as the ENTIRE population being examined.
Image
In fact, it helps more of you think of each blue pixel/dot under the curve as representing one member of the population.


So, if a score of 650 is at the 60th percentile, we know that 60% of all of the blue pixels/dots lie to the left of 650
Image


Likewise, if a score of 850 is at the 90th percentile, we know that 90% of all of the blue pixels/dots lie to the left of 850
Image



IMPORTANT: If we draw a new line that takes the population BETWEEN the 60th and 90th percentiles and divides that population into two EQUAL populations, then that new line will represent the 75th percentile score.
Image
So, where should that line go?

Well, if we draw it halfway between 650 and 850...
Image
...then more of the population will be on side A.
So, this line will NOT divide the population between the 60th and 90th percentiles into two equal populations.


To get a line the DOES divide the population into two equal populations, we need to draw it right about here...
Image


Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850)
Image

So, the SCORE associated with the 75th percentile is LESS THAN 750

Answer: B



Stunning. really...................
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Re: The random variable X is normally distributed [#permalink] New post 26 Apr 2020, 09:27
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Here the GRE gave us the midpoint 750 as a good bait but also as the problem solver. What if it gave a value skewed towards 650 and somewhere close to the actual value of the 75th percentile? How would we verify it then?
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Re: The random variable X is normally distributed [#permalink] New post 28 Apr 2020, 04:50
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I don't understand this?

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Re: The random variable X is normally distributed [#permalink] New post 29 Apr 2020, 13:08
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GreenlightTestPrep wrote:

Well, if we draw it halfway between 650 and 850...
Image
...then more of the population will be on side A.
So, this line will NOT divide the population between the 60th and 90th percentiles into two equal populations.


To get a line the DOES divide the population into two equal populations, we need to draw it right about here...
Image


Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850)
Image

So, the SCORE associated with the 75th percentile is LESS THAN 750

Answer: B


Hi Brent,

If 75 lies b/w 60 and 90 then why 750 doesn't lie in b/w 650 and 850?
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Re: The random variable X is normally distributed [#permalink] New post 30 Apr 2020, 01:32
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Farina wrote:
GreenlightTestPrep wrote:

Well, if we draw it halfway between 650 and 850...
Image
...then more of the population will be on side A.
So, this line will NOT divide the population between the 60th and 90th percentiles into two equal populations.


To get a line the DOES divide the population into two equal populations, we need to draw it right about here...
Image


Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850)
Image

So, the SCORE associated with the 75th percentile is LESS THAN 750

Answer: B


Hi Brent,

If 75 lies b/w 60 and 90 then why 750 doesn't lie in b/w 650 and 850?


I would like to answer your question by rephrasing your question

750 lies at the center of 650 and 850 but

75th percentile doesn't lie at the center of 60th and 90th percentile because Normal distribution is plotted using bell curve, where concentration of data decreases as you move away from the center

Between 60%le and 90%le you have 30% of values. 75%le would mean 15% of values i.e half of those values. As the concentration of data decreases away from the center , half of the values will be encountered before the actual mid of 650 and 850 i.e. 750.
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Re: The random variable X is normally distributed [#permalink] New post 30 Apr 2020, 01:36
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GreenlightTestPrep wrote:
sandy wrote:
The random variable X is normally distributed. The values 650 and 850 are at the 60th and 90th percentiles of the distribution of X, respectively.

Quantity A
Quantity B
The value at the 75th percentile of the distribution of X
750




VERY tricky question.

When it comes to normal distributions, it might be useful to think of the area under the curve as the ENTIRE population being examined.
Image
In fact, it helps more of you think of each blue pixel/dot under the curve as representing one member of the population.


So, if a score of 650 is at the 60th percentile, we know that 60% of all of the blue pixels/dots lie to the left of 650
Image


Likewise, if a score of 850 is at the 90th percentile, we know that 90% of all of the blue pixels/dots lie to the left of 850
Image



IMPORTANT: If we draw a new line that takes the population BETWEEN the 60th and 90th percentiles and divides that population into two EQUAL populations, then that new line will represent the 75th percentile score.
Image
So, where should that line go?

Well, if we draw it halfway between 650 and 850...
Image
...then more of the population will be on side A.
So, this line will NOT divide the population between the 60th and 90th percentiles into two equal populations.


To get a line the DOES divide the population into two equal populations, we need to draw it right about here...
Image


Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850)
Image

So, the SCORE associated with the 75th percentile is LESS THAN 750

Answer: B


Amazing explanation BRENT :wave
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Re: The random variable X is normally distributed [#permalink] New post 04 Jun 2020, 04:32
GreenlightTestPrep wrote:
sandy wrote:
The random variable X is normally distributed. The values 650 and 850 are at the 60th and 90th percentiles of the distribution of X, respectively.

Quantity A
Quantity B
The value at the 75th percentile of the distribution of X
750




VERY tricky question.

When it comes to normal distributions, it might be useful to think of the area under the curve as the ENTIRE population being examined.
Image
In fact, it helps more of you think of each blue pixel/dot under the curve as representing one member of the population.


So, if a score of 650 is at the 60th percentile, we know that 60% of all of the blue pixels/dots lie to the left of 650
Image


Likewise, if a score of 850 is at the 90th percentile, we know that 90% of all of the blue pixels/dots lie to the left of 850
Image



IMPORTANT: If we draw a new line that takes the population BETWEEN the 60th and 90th percentiles and divides that population into two EQUAL populations, then that new line will represent the 75th percentile score.
Image
So, where should that line go?

Well, if we draw it halfway between 650 and 850...
Image
...then more of the population will be on side A.
So, this line will NOT divide the population between the 60th and 90th percentiles into two equal populations.


To get a line the DOES divide the population into two equal populations, we need to draw it right about here...
Image


Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850)
Image

So, the SCORE associated with the 75th percentile is LESS THAN 750

Answer: B


Awesome Explanation.
Unfortunately, still I have following doubts.

How were you able to decide that 75%ile was not able to divide 60%lie and 90%ile in equal parts?
How are we able to determine the point which equally divide the graph between 60%ile and 90%ile equally?

Can someone give the basis of this??
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Re: The random variable X is normally distributed [#permalink] New post 04 Jun 2020, 05:16
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suramya26 wrote:

Awesome Explanation.
Unfortunately, still I have following doubts.

How were you able to decide that 75%ile was not able to divide 60%lie and 90%ile in equal parts?
How are we able to determine the point which equally divide the graph between 60%ile and 90%ile equally?

Can someone give the basis of this??


Q: How were you able to decide that 75%ile was not able to divide 60%lie and 90%ile in equal parts?
A: If the distribution who are in the shape of a rectangle, then each region would have the same area, which would mean a value of 750 would be in the 75th percentile. See below:
Image
Since distribution is not in the shape of a rectangle, we know that a value of 750 does not divide the two regions into equal areas.

Q: How are we able to determine the point which equally divide the graph between 60%ile and 90%ile equally?
A: That would require knowledge of z-scores, which is not tested on the GRE.

Cheers,
Brent
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Re: The random variable X is normally distributed   [#permalink] 04 Jun 2020, 05:16
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