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

\[\begin{array}{1 1}(1)120&(2)20\\(3)80&(4)160\end{array}\]

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Let X denotes the marks secured.
$X \sim N (65, \sigma)$
$Z =\large\frac{X-\mu}{\sigma} =\frac{X- 65}{\sigma}$
$P (X > 85) =\large\frac{120}{400}$
$P \bigg( Z > \large\frac{85-65}{\sigma} \bigg)= \frac{3}{10}$
$P \bigg( Z > \large\frac{20}{\sigma} \bigg)= \frac{3}{10}$-------(1)
$P (45 K< x< 65)$
$\qquad= P\bigg( \large\frac{45-65}{\sigma} < z < \large\frac{65-65}{\sigma}\bigg)$
$\qquad= P \bigg( -\large\frac{20}{\sigma} < z <0 \bigg)$
$\qquad= P \bigg(0 < z < \large\frac{20}{\sigma} \bigg)$
$\qquad =0.5 -P \bigg(z > \large\frac{20}{\sigma } \bigg)$
$\qquad= \large\frac{1}{2} -\frac{3}{10} $
$\qquad= \large\frac{5-3}{10} =\frac{2}{10} =\frac{1}{5}$
Number of student secured marks between 45 and 65 $ =\large\frac{1}{5} \times$$ 400$
Hence 3 is the correct answer.
answered May 23, 2014 by meena.p

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