Question

2. (15pts) The lengths of 100 railroad tracks segments are measured independently using the same device. The values obtained are assumed to be uniformly distributed between L - 10 millimeters and L10 millimeters, where L is the exact length of each one of them. Use the central limit theorem to approximate the probability that the overall estimated length of the railroad will be within 100 millimeters of the true length (the true length is 100L).

Answer 1

here error in one segment follow uniform distribution with parameter a=-10 and b=10

hence mean =(a+b)/2 =0

and std deviation =(b-a)/sqrt(12)=5.7735

for n=100; railroad segments ; expected deviation from true length =100*0 =0

and standard deviation of total error =5.7735*sqrt(100)=57.735

therefore from normal approximation;

P(overall estimated length will be within 100 milimeters of true length)=P(-100<X<100)

=P((-100-0)/57.735<Z<(100-0)/57.735)=P(-1.73<Z<1.73)=0.9582-0.0418 =0.9164