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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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Question: 5
Page: 156
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).


[Reveal] Spoiler: image
Attachment:
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Quant.jpg [ 22.35 KiB | Viewed 9244 times ]

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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:
[Reveal] Spoiler:
B

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Re: The random variable X is normally distributed [#permalink] New post 30 Jul 2017, 13:59
Nice explanation with graphics. Clear and clever.! Thanks.
Re: The random variable X is normally distributed   [#permalink] 30 Jul 2017, 13:59
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