Question

The math contest called the Fermat is out of 150 points. The mean score this year is 91 points with a standard deviation of 12. Students in the top 25% receive a medal. Assuming the scores are normally distributed, what mark is required to receive a medal? *

Answer 1

Given that:

$\mu=91$

$\sigma=12$

$n=150$

$\lambda=?$

$Area: 1-0.25=0.75$

The $Z-Score$ of $0.75$ is $0.67$.

$Z-score=\frac{\lambda -\mu }{\frac{\sigma }{\sqrt{n}}}$

$0.67=\frac{\lambda -91}{\frac{12}{\sqrt{150}}}$

$\lambda -91=0.67\times \frac{12}{\sqrt{150}}$

$\lambda =\left(0.67\times \frac{12}{\sqrt{150}}\right)+91$

$\lambda =91.6565\approx 91.66$

Therefore, $91.66$ are required to receive a medal.