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\title{Extensions on the Likelihood Ratio Test\footnote{\copyright 2019 Joseph C Watkins}}
\author{Worksheet 21}
\date{}
\maketitle

\begin{enumerate}
\item Snell's law \index{Snell's law} tell us how light bends at an interface - the angle of incidence versus the angle of refraction - based on the ratio of the velocities of light in the two isotropic media. If the angle of incidence of a laser beam in a vacuum is $\theta_1$ radians and the angle of refraction in an unknown medium is $\theta_2$, then 
$$n= \frac{\sin\theta_1}{\sin\theta_2}$$
is called the {\bf index of refraction}  ($\beta=1/n$ is the velocity of light in the medium as a fraction of the speed of light in a vacuum.) Make repeated independent measurements in radians, $\theta_{1,1}, \theta_{1,2}, \cdots, \theta_{1,16}$ of the angle of incidence in vacuum and $\theta_{2,1}, \theta_{2,2}, \cdots, \theta_{2,16}$, of the angle of refraction in the second medium. If these measurements have 
\begin{itemize}
\item sample mean $\overline{\theta}_1=0.786$ radians and standard deviation 0.03 radians in vacuum and
\item sample mean $\overline{\theta}_2=0. 326$ radians and standard deviation 0.06 radians in the unknown medium.
\end{itemize}
\begin{enumerate}
\item Snell's law gives an estimate $\hat n$ based on the values of  $\overline{\theta}_1$ and $\overline{\theta}_2$. Use the delta method to estimate the mean and standard deviation of $\hat n$.
\item You suspect that the substance is cubic zirconia ($n_z=2.165$) and not the claimed material, diamond ($n_d=2.418 $). For the hypothesis
$$H_0: n=n_d\quad\mbox{versus}\quad H_1:n=n_z,$$
use the information above to devise a $z$-test for the hypothesis and report a $p$-value for the test.
\item Describe what the $p$-value communicates in this case.
\end{enumerate}
\item Migrants are thought to be more likely than native-born Americans  to own a business,  
\begin{enumerate}
\item Give a hypothesis that is appropriate  to this situation. Define any population parameters that you use in stating the hypothesis\medskip

A 2012 study based on Current Population Survey (CPS) data from the U.S. Census Bureau, found that the existing business-ownership rate is higher for migrants than native-born adults in America: 10.5\% of the migrant workforce own a business, compared with 9.3\% of the native-born labor force.
\item Fill in the following table for the power for the given alternatives. Assume that 9.3\% of the native-born labor force  own a business, 

\begin{center}
\begin{tabular}{cccc}
migrant&significance&&number of \\
proportion&level&power&observations \\ \hline
0.105&0.05&0.90\\
0.105&0.01&0.95 \\
0.105&0.05&&1000 \\
0.105&0.05&&2000 \\
0.120&0.05&0.90 \\
0.120&0.01&0.95 \\
0.120&0.05&&1000 \\
0.120&0.05&&2000 \\ \hline
\end{tabular}
\end{center}
\end{enumerate}
\end{enumerate}

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\item One of the lenses in your supply is suspected to have a focal length $f$ of 9.1cm rather than the 9cm claimed by the manufacturer.
\begin{enumerate}
\item Write an appropriate hypothesis  test for this situation.
\item The focal length $f$ is determined by using the thin lens formula, \index{thin lens formula}
$$\frac{1}{s_1} + \frac{1}{s_2} = \frac{1}{f}.$$
Here $s_1$ is the distance from the lens to the object and $s_2$ is the distance from the lens to the real image of the object. The distances $s_1$ and $s_2$ are each independently measured 25 times. The sample mean of the measurements  is $\bar s_1 =26.6$ centimeters and $\bar s_2 = 13.8$ centimeters, respectively. The standard deviation of the measurement is 0.1cm for $s_1$ and 0.5cm for $s_2$.
\medskip

Give an estimate $\hat f$ based on these measurements and the thin lens formula.
\item Use the delta method to give the standard deviation of $\hat f$.
\item Use this to devise a $z$-test for the hypothesis and report a $p$-value for the test.
\end{enumerate}