# Marks in an aptitude test given to $800$ students of a school was found to be normally distributed. $10\%$ of the students scored below $40$ marks and $10\%$ of the students scored above $90$ marks. Find the number of students scored between $40$ and $90$.

Toolbox:
• Standard normal distribution:
• In a standard normal distribution $\mu=0,\sigma ^2=1$
• The random variable $X$ can be converted to the standard normal variable $Z$ by the transformation
• $Z=\large\frac{X-\mu}{\sigma}$
Step 1:
Let $X$ be the random variable denoting the marks scored by a student in a school with 800 students.
$P(X < 40)=0.1$
$P(X > 90)=0.1$
$\therefore P(40 < X < 90)=1-0.2=0.8$
Step 2:
The number of students who scored between 40 and 90=$800\times 0.8$
$\Rightarrow 640$