In: Math
40 numbers are rounded off to the nearest integer and then summed. If the individual round-off error are uniformly distributed over (−.5,.5) what is the probability that the resultant sum differs from the exact sum by more than 2 ?
Let the error of individual round off be denoted by Wi.
Here
If the original numbers are Xi and the numbers to which they are rounded off are Yi,
then, Xi=Yi + Wi.
It is given that
Hence
Hence by Central limit theorem
The exact sum is given by
And sum of rounded off numbers is given by
Therefore the sum of errors will be given by
Note that
We want the probability that the resultant sum differs from the exact sum by more than 2.
Which is
P[The resultant sum differs from the exact sum by more than 2]
P[The resultant sum differs from the exact sum by more than 2] = 0.2734