Q.The weight of randomly selected 500 adult persons from a region of a city follows normal distribution. The average weight of these persons in 55 kg and its standard deviation is 7 kg:

$Z_{41}=-2$, $Z_{62}=1$. (i) Area $=0.4772+0.3413=0.8185\Rightarrow 0.8185\times500\approx\mathbf{409}$ persons. (ii) $P(Z<-2)=0.0228=\mathbf{2.28\%}$.

Given: weights $\sim$ Normal, $\mu=55$ kg, $\sigma=7$ kg, $N=500$ persons.

Standardise: $Z=\dfrac{x-\mu}{\sigma}$.

(i) Between 41 kg and 62 kg.

$Z_1=\frac{41-55}{7}=\frac{-14}{7}=-2,\qquad Z_2=\frac{62-55}{7}=\frac{7}{7}=1.$

Required area $=P(-2<Z<1)=P(0<Z<2)+P(0<Z<1)=0.4772+0.3413=0.8185$.

Number of persons:

$0.8185\times 500=409.25\approx 409\ \text{persons}.$

…