Question

The Scholastic Aptitude Test (SAT) scores in mathematics at a certain high school are normally distributed, with a mean of 450 and a standard deviation of 100. What is the probability that an individual chosen at random has the following scores? (Round your answers to four decimal places.)

(a) greater than 650
(b) less than 250
(c) between 500 and 550

Answer 1

We know that a Normal random variable with mean $\mu$ and standard deviation $\sigma$ , the CDF of the random variable would be

$F(x) = \Phi\left ( \frac{x-\mu}{\sigma} \right )$

Where $\Phi$ is the CDF of the standard normal random variable.

In this case, let us assume that X is the SAT score of an individual. Then, we get that the probability of getting a score

a) Greater than 650 is

$= P(X \ge 650) \\ = 1- F_X(650) \\ = 1-\Phi\left ( \frac{650-450}{100} \right ) \\ = 1 - \Phi(2) = 0.0228$

b) less than 250 is

$= P(X \le 250) \\ = F_X(250) \\ = \Phi\left ( \frac{250-450}{100} \right ) \\ = \Phi(-2) = 0.0228$

c) between 500 and 550 is

$= P(550\ge X \ge 500) \\ = F_X(550) - F_X(500) \\ = \Phi\left ( \frac{550-450}{100} \right )- \Phi\left ( \frac{500-450}{100} \right ) \\ = \Phi(1)-\Phi(0.5) = 0.1499$

The values for the CDF of standard normal can be found in the normal distribution table.