The weights of 1,000 children, in average, is 50kg with standard deviation of 14kg. Suppose the weights are normally distributed, how many children weigh between 53kg and 68kg?

Let X be a random variable represents the weight of randomly selected child, then X~$N(50,14^2)$

$P(53<X<68)=P(53<N(50,14^2)<68)=P(53<50+14N(0,1)<68)=P(0.214<N(0,1)<1.071)=P(N(0,1)<1.071)-P(N(0,1)<0.214)=0.85792-0.58473=0.27319$

So, approximately 1000*0.27319=273.19=(round to integer)=273 children have such a weight.