The length of a component is to be estimated through

QUESTION:

The length of a component is to be estimated through repeated measurement.
a. Ten independent measurements are made with an instrument whose uncertainty is . Let \(\overline X\) denote the average of these measurements. Find the uncertainty in \(\overline X\).

b. A new measuring device, whose uncertainty is only , becomes available. Five independent measurements are made with this device. Let \(\overline Y\) denote the average of these measurements. Find the uncertainty in \(\overline Y\).

c. In order to decrease the uncertainty still further, it is decided to combine the estimates \(\overline X\) and \(\overline Y\). One engineer suggests estimating the length with \((1/2)\overline X+(1/2)\overline Y\). A second engineer argues that since \(\overline X\) is based on 10 measurements, while \(\overline Y\) is based on only five, a better estimate is \((10/15)\overline X+(5/15)\overline Y\). Find the uncertainty in each of these estimates. Which is smaller?

d. Find the value  such that the weighted average \(c\overline X+(1-c)\overline Y\) has minimum uncertainty. Find the uncertainty in this weighted average.

Equation Transcription:

Text Transcription:

overline{X}

overline{X}

overline{Y}

overline{Y}

overline{X}

overline{Y}

(1/2)overline{X}+(1/2)overline{Y}

overline{X}

overline{Y}

(10/15)overline{X}+(5/15)Y

c overline{X}+(1-c)overline{Y}