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The random variable X is normally distributed [#permalink]
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Question Stats: 36% (00:55) correct 63% (00:51) wrong based on 141 sessions
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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Question: 5
Page: 156
Difficulty: hard
[Reveal] Spoiler: OA

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Sandy
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Re: The random variable X is normally distributed [#permalink]
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Now this is a question that essentially tests our understanding of the bell curve. 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).

[Reveal] Spoiler: image
Attachment: Quant.jpg [ 22.35 KiB | Viewed 40119 times ]

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Re: The random variable X is normally distributed [#permalink]
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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]
Expert's post
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]
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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. 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 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 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. So, where should that line go?

Well, if we draw it halfway between 650 and 850... ...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... Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850) So, the SCORE associated with the 75th percentile is LESS THAN 750

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Brent Hanneson – Creator of greenlighttestprep.com

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Re: The random variable X is normally distributed [#permalink]
Nice explanation with graphics. Clear and clever.! Thanks. Manager Joined: 27 Feb 2017
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Re: The random variable X is normally distributed [#permalink]
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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? Retired Moderator Joined: 07 Jun 2014
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Re: The random variable X is normally distributed [#permalink]
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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]
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Very nice explanation Manager  Joined: 01 Nov 2018
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Re: The random variable X is normally distributed [#permalink]
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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. In fact, it helps more of you think of each blue pixel/dot under the curve as representing one member of the population.

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 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 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. So, where should that line go?

Well, if we draw it halfway between 650 and 850... ...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... Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850) So, the SCORE associated with the 75th percentile is LESS THAN 750

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Re: The random variable X is normally distributed [#permalink]
Expert's post
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. 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 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 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. So, where should that line go?

Well, if we draw it halfway between 650 and 850... ...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... Notice that the 75th percentile line is to the left of score of 750 (which is halfway between scores of 650 and 850) So, the SCORE associated with the 75th percentile is LESS THAN 750

Stunning. really...................
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