The length of a component is to be estimated through repeated measurement.
a. Ten independent measurements are made with an instrument whose uncertainty is 0.05 mm. Let X denote the average of these measurements. Find the uncertainty in X.
b. A new measuring device, whose uncertainty is only 0.02 mm, becomes available. Five independent measurements are made with this device. Let Y denote the average of these measurements. Find the uncertainty in Y.
c. In order to decrease the uncertainty still further, it is decided to combine the estimates X and Y. One engineer suggests estimating the length with (1/2)X + (1/2)Y . A second engineer argues that since X is based on 10 measurements, while Y is based on only five, a better estimate is (10/15)X +(5/15)Y . Find the uncertainty in each of these estimates. Which is smaller?
d. Find the value c such that the weighted average cX +(1−c)Y has minimum uncertainty. Find the uncertainty in this weighted average.