Question

When a randomly selected number A is rounded off to the nearest integer R_A, it is reasonable to assume that the round-off error A- R_A is uniformly distributed on the interval (-.5,.5)

If 50 numbers are rounded off to the nearest integer and then averaged, approximate the probability that the resulting average differs from the exact average of the 50 numbers by more than 0.1 (round your answer to 4 digits).

Answer 1

here for uniform distribution parameter a =-0.5 and b=0.5

mean μ=(a+b)/2 =	0
varaince =σ2= (b-a)2/12 =		0.0833
standard deviation σ=(b-a)/√12=		0.2887

since n=50 is greater than 30 , from CLT we can use normal approximation of sampling mean

for normal distribution z score =(X-μ)/σ
here mean=       μ=	0
std deviation   =σ=	0.289
sample size       =n=	50
std error=σx̅=σ/√n=	0.04082

  probability that the resulting average differs from the exact average of the 50 numbers by more than 0.1:

probability =1-P(-0.1<X<0.1)=1-P((-0.1-0)/0.041)<Z<(0.1-0)/0.041)=1-P(-2.45<Z<2.45)=1-(0.9929-0.0071) =0.0142