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\title{Composite Hypotheses\footnote{\copyright 2018 Joseph C Watkins}}
\author{Worksheet 20}
\date{}
\maketitle



\begin{enumerate}\item In the shell game, three identical shells are placed face-down on a surface. A small ball is placed beneath one of these shells so that it cannot be seen, and the shells are then shuffled by the operator in plain view. 

\begin{enumerate}
\item Give an appropriate hypothesis for the case that the player choice for the shall is better than chance.
\item Out of playing the game 60 times,  the guess is correct 25 times.   Test the hypothesis based on the binomial distribution and report a $p$-value.
\item Is the test significant at the 10\%,  5\%, 1\% level?
\item Does your conclusion change if the guess is correct 27 times? 29 times?
\end{enumerate}
\item Younger Americans are better than their elders at separating factual from opinion statements in the news, according to a new analysis from Pew Research Center. Their analysis is based on a nationally representative survey of 5,035 adults 18 years of age or older conducted from February 22 to March 4, 2018.
\begin{enumerate}
\item The United States which has 15\% of the population between 18 and 29 years old and  22\% between 50 and 64 year old. Estimate the sample sizes for each of these two age groups.
\item For the younger age group, 34\% were able to correctly classify 5 factual and 5 opinion statements.  For the older group, 22\% made the classification correctly. Create 95\% confidence intervals for the  proportion of Americans of the appropriate age who would make the correct classification.
\item  An educational program aimed at the youth is designed to help improve the ability to classify a fact from an opinion. With a hypothesis
$$H_0: p\le p_0\quad\mbox{\rm versus}\quad H_1:p>p_0,$$
 and for $p_0=0.34$, give the value of the power function $\pi(p)$ for $p = 0.34, 0.35,0.36,0.37.0.38$ and $0.39$  with the choice of $\alpha = 0.02$ and  a sample of size 750.
\item  What qualititative change would you see in the  power curve change if $\alpha$ is reduced to 0.01? Explain your answer.
\item Compute the power function for $\alpha=0.02$  for a sample size of 1,500 for the same values for $p$
\item Give a sketch of the power curves in (c) and (e) and explain the difference in the two power functions.
\end{enumerate}
\end{enumerate}
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}\item  In a study of human memory, Shachs (1967) demonstrated that people recall the meaning of verbal material, but tend to forget the exact word-for-word details.  In this study, people read a passage of text.  Then the people were shown a test sentence and asked whether or not the identical sentence has appeared in the text.  In one condition, the test sentence was phrased differently, but had the same meaning as a sentence that was in the text.   \index{human memory}
\begin{enumerate}
\item Write a hypothesis to test that the performance is significantly better than expected by chance.
\item Out of 45 people, 27 correctly noticed that change in the sentence. Test the hypothesis based on the binomial distribution and report a $p$-value.
\item Is the test significant at the 10\%,  5\%, 1\% level?
\item Does your conclusion change is 28 people notice the change? 29 people?
\end{enumerate}
\item Younger Americans are better than their elders at separating factual from opinion statements in the news, according to a new analysis from Pew Research Center. Their analysis is based on a nationally representative survey of 5,035 adults 18 years of age or older conducted from February 22 to March 4, 2018.
\begin{enumerate}