Question

The marks secured by $400$ students in a Mathematics test were normally distributed with mean $65$. If $120$ students got marks above $85$, the number of students securing marks between $45$ and $65$ is

• A. $120$
• B. $20$
• C. $80$
• D. $160$

Answer 1

The correct option is C $80$

Let $X$ denote the marks secured.

Given, $\mu =65$

Thus, $X\sim N(65,\rho )$

$\Rightarrow z=\frac{X-\mu }{\rho }=\frac{X-65}{\rho }$

$\Rightarrow P(X>85)=\frac{120}{400}$

$\Rightarrow P(z>\frac{85-65}{\rho })=\frac{3}{10}$

$\Rightarrow P(z>\frac{20}{\rho })=\frac{3}{10}$ ....(1)

$\Rightarrow P(45<x<65)$ $=P(\frac{45-65}{\rho }<z<\frac{65-65}{\rho })$

$=P(\frac{-20}{\rho }<z<0)$

$=P(0<z<\frac{20}{\rho })$

$=0.5-P(z>\frac{20}{\rho })$

$=\frac{1}{2}-\frac{3}{10}$

$=\frac{1}{5}$

Number of students secured marks between $45$ and $65$ $=\frac{1}{5}\times 400=80$.

Hence, the correct answer is option .