lmcelduff wrote:Hello, I was working through a GMAT Math CAT and could not solve this problem:
The members of the newest recruiting class of a certain military organization are taking their physical conditioning test, and those who score in the bottom 16 percent will have to retest. If the scores are normally distributed and have an arithmetic mean of 72, what is the score at or below which the recruits will have to retest?
(1) There are 500 recruits in the class.
(2) 10 recruits scored 82 or higher.
Help would be greatly appreciated!!
A normal distribution is a bell-shaped curve, symmetrical about its mean. You can read more about it here:
https://mathworld.wolfram.com/NormalDistribution.html
So imagine a frequency distribution plot, showing frequency (count of students) on the y-axis and score on the x-axis. The question stimulus asks the score at or blow which the recruits will have to retest. In other words, its asking the point on the x-axis to the left of which 16 % of students lie.
To be able to answer this, the distribution curve must be known completely. i.e. you should not only know the height of the curve, but also the width of the curve (typically characterized by FWHM - Full Width at Half Maximum)
Mean = 72 => the bell curve is symmetrical about x=72.
1. There are a total of 500 students.
From this information alone you cannot "fix" the curve. For example, you could have 400 students at score of 72, and 50 each at score of 70 and 74. In such a case the bell would be very narrow and sharp looking - high at the mean and narrow width.
You could also have a broad-looking curve -> 200 at 72, 100 each at 60 and 84, and 50 each at 50 and 94.
In other words, statement 1 says that area under the curve is 500, but does not give any information on the width or height of the curve. The curve can be drawn in several ways, hence INSUFFICIENT.
2. 10 recruits scored 82 or higher.
From this information alone, a fixed curve cannot be drawn. Since the total number of students is not known, one could a curve such that 10 have 82 or higher score, any number of students at the mean value of 72. It could be a 100 or 1000. i.e. with this information, the height of the curve cannot be fixed. Hence INSUFFICIENT.
Combine the two ->
Now, parameters required to fix the curve are known.
Area under the curve = 500. The mean = 72. A point on the right of the mean through which the curve passes. (1) and (2) combined are sufficient to fix the curve and hence answer the question.
Answer C.
However, this question is outside the GMAT scope and would never appear on the test.
