Question

What is the mean of the round-off errors when measuring the distance that a long jumper has jumped, which is uniformly distributed between 0 and 4.6 mm and rounded to four decimal places when possible?

Answer 1

The mean of the round-off errors for measurements uniformly distributed between 0 and 4.6 mm, rounded to four decimal places, is 0.00005 mm.

Step-by-step explanation:

The question asks about the mean of the round-off errors when measuring the distance of a long jump that is uniformly distributed between 0 and 4.6 mm and rounded to four decimal places when possible. To compute this, we must understand that the round-off error for a uniformly distributed measurement is the expected value of that error over the range of measurements. Because the distribution is uniform, every error between the smallest possible error (0 mm, if no rounding is needed) and the largest possible error (just under the smallest increment that would cause rounding up, which is 0.0001 mm due to rounding to four decimal places) is equally likely.

In this case, because the errors are distributed from 0 to 0.0001 mm, the mean error is simply the average of these two values. So the mean round-off error would be (0 + 0.0001)/2 = 0.00005 mm. This represents the average discrepancy one would expect when rounding these measurements of jump distances to the nearest ten-thousandth of a millimeter.