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\title{Central Limit Theorem\footnote{\copyright 2019 Joseph C Watkins}}
\author{Worksheet 14}
\date{}
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\begin{enumerate}
\item The focal length $f$ of an optical instrument is needed. This is determined by using the thin lens formula,
$$\frac{1}{r_1} + \frac{1}{r_2} = \frac{1}{f}.$$ 
where $r_1$ is the distance from the lens to the object and $r_2$ is the distance from the lens to the real image of the object.
The distance $r_1$ is independently measured 12 times and $r_2$ is independently measured 40 times. The mean of the measurements is the actual distances, 9 centimeters and 18 centimeters, respectively. The standard deviation of the measurement is 0.1 centimeters for $r_1$ and 0.5 centimeter for $r_2$.
\begin{enumerate}
\item Let $\bar R_1$ be the sample mean of the 12 measurements to the object. Estimate, using the central limit theorem, $P\{\bar R_1 < 8.9 \hbox{cm}\}$.
\item Let $\bar R_2$ be the sample mean of the 40 measurements to the image. Estimate, using the central limit theorem, $P\{\bar R_2 < 17.9 \hbox{cm}\}$.
\item How many measurements are needed so that  $P\{|\bar R_2 -18\hbox{cm}|>0.1\hbox{cm}\}\le 0.02$.
\item Would more or fewer measurements be needed so that 
\begin{enumerate}
\item $P\{|\bar R_2 -18\hbox{cm}|>0.1\hbox{cm}\}\le 0.01$?
\item $P\{|\bar R_2 -18\hbox{cm}|>0.2\hbox{cm}\}\le 0.02$?
\end{enumerate}
Explain your answer.
\item For measurements $r_{1,1},\ldots, r_{1,12}$ and $r_{2,1},\ldots, r_{2,40}$, estimate the focal length using 
$$\frac{1}{\bar r_1} + \frac{1}{\bar r_2} = \frac{1}{\hat f}.$$
Use the delta method to give an estimate of the standard deviation of $\hat f$.
\item Simulate this protocol 10000 times using the {\tt rnorm} command and compare the results from your simulation to the results from the delta method.
\end{enumerate}\end{enumerate}

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