sandy wrote:
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.
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
_________________
Brent Hanneson - founder of Greenlight Test Prep