[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

A surveyor is measuring the height of a cliff known to be about 1000

feet. He assumes his instrument is properly calibrated and that his measurement errors are independent, with mean [math]\mu = 0[/math] and variance [math]\sigma^2 = 10[/math]. He plans to take [math]n[/math] measurements and form the average. Estimate, using (a) Chebyshev's inequality and (b) the normal approximation, how large [math]n[/math] should be if he wants to be 95 percent sure that his average falls within 1 foot of the true value. Now estimate, using (a) and (b), what value should [math]\sigma^2[/math] have if he wants to make only 10 measurements with the same confidence?