The length of a component is to be estimated through repeated measurement. a. Ten

Chapter 3, Problem 19

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The length of a component is to be estimated through repeated measurement.

a. Ten independent measurements are made with an instrument whose uncertainty is \(0.05 \mathrm{~mm}\). Let \(\bar{X}\) denote the average of these measurements. Find the uncertainty in \(\bar{X}\).

b. A new measuring device, whose uncertainty is only \(0.02 \mathrm{~mm}\), becomes available. Five independent measurements are made with this device. Let \(bar{Y}\) denote the average of these measurements. Find the uncertainty in \(bar{Y}\).

c. In order to decrease the uncertainty still further, it is decided to combine the estimates

\(\bar{X} \text { and } \bar{Y}\). One engineer suggests estimating the length with (1 / 2) \(\bar{X}^{-}+(1 / 2) \bar{Y}\). A second engineer argues that since \(\bar{X}\) is based on 10 measurements, while \(\bar{Y}\) is based on only five, a better estimate is \((10 / 15) \bar{X}+(5 / 15) \bar{Y}\). Find the uncertainty in each of these estimates. Which is smaller?

d. Find the value  such that the weighted average \(c \bar{X}+(1-c) \bar{Y}\) has minimum uncertainty.

 Find the uncertainty in this weighted average.

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Text Transcription:

0.05 mm

\bar{X}

\bar{X}

0.02 mm

\bar{Y}

\bar{Y}

\bar{X} \text { and } \bar{Y}

(1 / 2) \bar{X}^{-}+(1 / 2) \bar{Y}

\bar{X}

\bar{Y}

(10 / 15) \bar{X}+(5 / 15) \bar{Y}

c \bar{X}+(1-c) \bar{Y}