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QOTD#13 The figure above shows the standard normal

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QOTD#13 The figure above shows the standard normal [#permalink] New post 19 Aug 2016, 15:58
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The figure above shows the standard normal distribution, with mean 0 and standard deviation 1, including approximate percents of the distribution corresponding to the six regions shown.

The random variable Y is normally distributed with a mean of 470, and the value Y = 340 is at the 15th percentile of the distribution. Of the following, which is the best estimate of the standard deviation of the distribution?

A. 125
B. 135
C. 145
D. 155
E. 165

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Re: QOTD#13 The figure above shows the standard normal [#permalink] New post 19 Aug 2016, 16:01
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Explanation

Since you know that the distribution of the random variable Y is normal with a mean of 470 and that the value 340 is at the 15th percentile of the distribution, you can estimate the standard deviation of the distribution of Y using the standard normal distribution. You can do this because the percent distributions of all normal distributions are the same in the following respect: The percentiles of every normal distribution are related to its standard deviation in exactly the same way as the percentiles of the standard normal distribution are related to its standard deviation.

For example, approximately 14% of every normal distribution is between 1 and 2 standard deviations above the mean, just as the figure above illustrates for the standard normal distribution.

From the figure, approximately 2% + 14%, or 16%, of the standard normal distribution is less than –1. Since 15% < 16%, the 15th percentile of the distribution is at a value slightly below –1. For the standard normal distribution, the value –1 represents 1 standard deviation below the mean of 0. You can conclude that the 15th percentile of every normal distribution is at a value slightly below 1 standard deviation below the mean.

For the normal distribution of Y, the 15th percentile is 340, which is slightly below 1 standard deviation below the mean of 470. Consequently, the difference 470 – 340, or 130, is a little greater than 1 standard deviation of Y; that is, the standard deviation of Y is a little less than 130. Of the answer choices given, the best estimate is 125, since it is close to, but a little less than, 130. The correct answer is Choice A.
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Re: QOTD#13 The figure above shows the standard normal [#permalink] New post 02 May 2017, 12:35
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sandy wrote:
Image

The figure above shows the standard normal distribution, with mean 0 and standard deviation 1, including approximate percents of the distribution corresponding to the six regions shown.

The random variable Y is normally distributed with a mean of 470, and the value Y = 340 is at the 15th percentile of the distribution. Of the following, which is the best estimate of the standard deviation of the distribution?

A. 125
B. 135
C. 145
D. 155
E. 165


Tricky!!!

First recognize that about 2% of the entire population is MORE THAN 2 units of standard deviation less than the mean.
In other words, if a value is 2 units of standard deviation below the mean, then that value is at the 2nd percentile.
Image

Also recognize that about 16% of the entire population is MORE THAN 1 unit of standard deviation less than the mean.
In other words, if a value is 1 unit of standard deviation below the mean, then that value is at the 16th percentile.
Image

So, a value that's at the 15th percentile will be a little more than 1 unit of standard deviation less than the mean.
Image

We're told that a certain normally distribution has a mean of 470.
So, it will look like this:
Image

We're also told that 340 is at the 15th percentile of the distribution. So, we'll add that here:
Image
Notice that the value associated with the 15th percentile is MORE THAN 1 standard deviation below the mean.

If we had to approximate, we might say that 340 is 1.1 unit of standard deviation below the mean.
NOTE: This approximation is not that important, as long as we recognize that 340 is MORE THAN 1 standard deviation below the mean. You'll see why this is shortly.

The mean = 470, which means 340 is 130 less than the mean
In other words, 130 ≈ 1.1 unit of standard deviation
Or we can write: 130 ≈ (1.1)(the standard deviation of the population)
If we solve this we get: the standard deviation of the population ≈ 130/1.1

VERY IMPORTANT: What really matters here is that the value of 130/1.1 is LESS THAN 130, which means the correct answer must be A, since there's only one answer choice that is less than 130.

Cheers,
Brent
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Re: QOTD#13 The figure above shows the standard normal [#permalink] New post 14 Apr 2018, 09:02
But 16% of the data is also 1 SD above the mean ... So a value a little more than 130 can also be an answer ... I think the answer could be A or B
Re: QOTD#13 The figure above shows the standard normal   [#permalink] 14 Apr 2018, 09:02
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