Suppose we have a set $A$ of $84965$ integers. The distribution of $A$ is supposed to be a normal distribution with Expectation of $500$ and Standard Deviation of $100$ theoretically. However, it is just a theoretical assumption, wether it is indeed a normal distribution is unknown.
Information we already know:
Out of the $84965$ integers, $120$ of them are greater than or equal to $800$, $2017$ of them are greater than or equal to $700$, and $14018$ of them are greater than or equal to $600$.
The question is:
If we know a person's College Entrance Exams grade is $722$, can we estimate the person's ranking according to this grade? In other words, can we estimate how many of the $84965$ integers are greater than or equal to $722$?
And please show your reasoning of estimation.
B.t.w., my estimation of the ranking is about $1100$ - $1200$, is it right?
Let's check if it is indeed the theoretical distribution:
So it is a fair assumption that A follows that distribution. Now let's apply a common transform: since $A \sim N(500, 100), Z = \frac{A - 500}{100}$ follows a standard normal distribution, $Z \sim N(0,1)$. For $A = 500$ we need $\phi (2,22)$ which can be found in precalculated tables about $N(0,1)$. Given that the probability that a person's grade is $722$ is now known, I will let you finish the estimation.