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.


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.


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


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


We're also told that 340 is at the 15th percentile of the distribution. So, we'll add that here:

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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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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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

the question can be rephrased as : what is 1 unit of sd ? so the given sd at 15%, which is greater than 1sd and so this obviously mean that 1sd is lower

I think, per question, it is not above but below the mean. Hence, 130 is not a correct answer.

130/1,1=117,2 if there was something else< for example 120, it could be difficult to realize what is true 120 or 125

The simple approach to take here is to consider that 2% is 2 S.D below the mean. That is 2nd percentile is 2 S.D below the mean. Now if we add up the 14%, that becomes 16% of the data is 1 S.D below the mean and we are given that Y=340 is at the 15th percentile that is 15%. So 15% is 340 and it is 1 S.D below the mean so that S.D should be somewhere around 130. Now it is the 15th percentile and the exact 1 S.D is at 16%, so we move towards the mean and so it will be greater than 370, so the S.D is less than 130 and so 125.

Standard deviation shows the dispersion of data. the more the data is spread, then higher is the standard deviation from the mean. therefore, if data is far from the mean, then the more is the standard deviation value.
from the question, is is cleared that the value at 15 percentile is less than the value at 16 percentile because as we move from left to right, the value Y increases. therefore, standard deviation decreases.
value at 15th percentile= 340, then standard deviation= 470-340= 130
let us suppose the value Y after 15th percentile (moving from left to right, it increases) at 16th percentile= 341
then standard deviation (which is the required one)= 470-341= 129
therefore the value of standard deviation should be less than 130, and the answer near to this is 125.
Answer= A