A student estimates from with , (Gaussian)

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Data analysis and model fitting in the laboratory

Measurement and Method, Ch Monte Carlo and the bootstrap: uncertainty by sampling

Problem

A student estimates

g

from

g = 4\pi^2 L/T^2

with

L = 1.000\,\mathrm{m}

u(L) = 0.005\,\mathrm{m}

(Gaussian), and a period timed by hand,

T = 2.00\,\mathrm{s}

u(T) = 0.30\,\mathrm{s}

(Gaussian, from the scatter of a few timings). The two inputs are independent. ((a)) Use the first-order rule to report

g

with its standard uncertainty. ((b)) Design a simulation that checks (a): say what each row of a spreadsheet holds, how each input is drawn, and what is recorded. Say how many trials the

1/\sqrt{2B}

rule asks for to know a standard deviation to

1

per cent of itself, and why it is prudent to run more here. ((c)) A run of

B = 10\,000

trials, sorted, has its

15.9

th,

50

th and

84.1

th percentiles at

7.50

9.88

and

13.72\,\mathrm{m\,s^{-2}}

. Report

g

about the best estimate from (a) as an asymmetric

68

per cent interval, and say whether the first-order interval describes it fairly. ((d)) Suppose the stopwatch in fact started a fixed

0.1\,\mathrm{s}

late on every timing, and nobody knew. Could any number of trials of your simulation reveal this? What could?

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