The GRE is widely used to help predict the performance of applicants to graduate schools. The range of possible scores on a GRE is 200 to 900. The psychology department at a university finds that the students in their department have scores with a mean of 544 and standard deviation of 103.

a. Find the probability that a student in the psychology department has a score less than 480.

b. Find the probability that a student in the psychology department has a score between 480 and 730.

#### Solution

Let $X$ denote the GRE Score. Given that the psychology department at a university finds that the students in their department have scores with a mean of 544 and standard deviation of 103. That is $\mu = 544$ and $\sigma = 103$.

Assuming that the GRE scores follows normal distribution with mean $\mu =544$ and standard deviation $\sigma = 103$. That is $X\sim N(544, 103^2)$.

Then $Z = \dfrac{X-\mu}{\sigma}$ follows standard normal distribution $N(0,1)$.

a. The probability that a student in the psychology department has a score less than $480$ is

`$$ \begin{aligned} P(X< 480)&=P\bigg(\frac{X-\mu}{\sigma} < \frac{480-544}{103}\bigg)\\ &= P(Z < -0.621)\\ &=0.2672 \end{aligned} $$`

b. The probability that a student in the psychology department has a score between $480$ and $730$ is

`$$ \begin{aligned} P(480 \leq X\leq 730) &=P(480 \leq X\leq 730)\\ &=P\bigg(\frac{480-544}{103}\leq \frac{X-\mu}{\sigma} \leq \frac{730-544}{103}\bigg)\\ &=P\bigg(-0.621 \leq Z \leq 1.806\bigg)\\ &= P(Z < 1.806) -P(Z < -0.621)\\ &=0.9645-0.2672\\ &= 0.6973 \end{aligned} $$`