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 Post subject: GMAT Statistics
PostPosted: Fri May 03, 2013 9:50 am 
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In a normal distribution, 68 percent of all values lie within one standard deviation of the mean, and 95 percent of the values lie within two standard deviations of the mean. If the mean of a symmetrical normally distributed set of values is 40 and the standard deviation is 10, what percent of all the values in the distribution lie between 50 and 60?
A. 2.5%
B. 13.5%
C. 20%
D. 34%
E. 95%

(B) A normal distribution is perfectly symmetric about the mean. Therefore, if 68% of the values are within one standard deviation of the mean and 95% are within two standard deviations, then 95% – 68% = 27% are more than 1 standard deviation from the mean but less than two.

Half of this 27%, which is 13.5%, will lie between 1 and 2 standard deviations below the mean, and the other 13.5% will lie between 1 and 2 standard deviations above the mean. The values above 50 and less than 60 are more than 1 standard deviation above the mean, but less than two.

The correct answer is choice (B).
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I think you have a mistake in this question. Within one standard deviation would be between 35 and 45, since the standard deviation is 10. Within 2 standard deviations would be between 30 and 50... therefore between 50 and 60 are in the 5%... so it would be aprox 2.5% (1/2 at each side of the bell). Do you agree?


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 Post subject: Re: GMAT Statistics
PostPosted: Fri May 03, 2013 9:54 am 
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The phrase in the question statement
Quote:
... within one standard deviation of the mean ...
tells us about a distance. It defines all data points, a distance from which to the mean does NOT exceed one standard deviation.

On a number line such statement defines an interval [base point – distance, base point + distance]:
Image

So in our case, we get the interval [meanstandard deviation, mean + standard deviation], which is [40 – 10, 40 + 10] = [30, 50].

So the proper interval is [30, 50], NOT [35, 45], which would be right, if the statement was "... within 5 of the mean ...".

P.S.
On a coordinate plane such statement, "... within distance of the base point ..." defines a circle
Image
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 Post subject: Re: GMAT Statistics
PostPosted: Fri May 03, 2013 9:55 am 
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I still don't get it, can you explain it using an example please?
Thank you.


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 Post subject: Re: GMAT Statistics
PostPosted: Fri May 03, 2013 9:56 am 
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The following statement in your reasoning:
Quote:
Within one standard deviation would be between 35 and 45, since the standard deviation is 10.
is NOT correct.
So let's concentrate on what the phrase
Quote:
68 percent of all values lie within one standard deviation of the mean
means.

The structure of this phrase is:
some set of values lie within x of y.

EXAMPLE: "Half of all values lie within 5 units of 10". It sets up the range [10 – 5, 10 + 5]. So this half of the values lie in the range [5, 15] (which means between 5 and 15).

In other words, phrase "lie within 5 units of 10" means "no farther than 5 units from 10" and it defines the range [5, 15], which is 5 × 2 = 10 units wide.
------------------------------------

Also NOTE, that reasoning
Quote:
therefore between 50 and 60 are in the 5%
could NOT be made based on the fact that 95 percent of the values lied within two standard deviations of the mean, even if the area within two standard deviations was from 30 to 50.

That fact, 95 percent of the values lie within two standard deviations of the mean, would have implemented that 5% lie outside the range [30, 50]. That range is NOT limited by 60 or any other number. So there would have been less than 2,5% of the values, which had lied from 50 to 60.


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