Problem

The Scholastic Aptitude Test (SAT) is rescaled so that the scores of n people, ranging from 0 to 1600, fit a distribution in the shape of the following function:

What is the probability that a random person will score between 1200 and 1250?

Solution

The probability we seek is a fraction equal to the area under the curve between x = 1200 and x = 1250, divided by the area under the curve between x = 0 and x = 1600. This is true because the area under the curve between two scores represents the number of people who achieved scores in that range.

Let's graph f(x), where x is between 0 and 1600, for n = 2 million (the approximate number of people who took the SAT in 2000–2001). The area shaded in magenta is the number of people with scores between 1200 and 1250.

The first integral we need to compute is

To solve the integral, notice that we can factor out np/3200, which is constant, to get

Since the derivative of a negative cosine function is a positive sine function, we might try something like

as an antiderivative. To check this, compute F'(x), using the chain rule:

We get an extra factor of p/1600 in F'(x), so we must divide the original F(x) by this factor.

Now the Fundamental Theorem of Calculus allows us to evaluate the integral.

This equals

Since this is the number of people with scores between 1200 and 1250, the percentage of people achieving such scores is simply

as long as the area under the entire curve equals n. Let's check:

This simplifies to