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{\Large \textbf{STA 312f22 Assignment Four}}\footnote{Copyright information is at the end of the last page.}
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\noindent
Please bring hard copy of your complete R input and output from Question \ref{Rquestion} to the quiz. The non-computer questions are practice for the quiz on Friday Oct. 21st, and are not to be handed in.  

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\hrule

\begin{enumerate} 


    \item This is a better version of a question from Assignment 3. Based on a random sample of size $n$ from a Bernoulli, you want to test $H_0: \pi=\pi_0$.
        \begin{enumerate}
            \item Write the formula for the likelihood ratio statistic $G^2$ as a function of $n$, $p$ and $\pi_0$.
            \item  Write the formula for the Pearson chi-squared statistic $X^2$ as a function of $n$, $p$ and $\pi_0$. Keep simplifying until it equals $Z_1^2$, where $Z_1$ is given on the formula sheet.
        \end{enumerate} 

    \item What do you call a study in which the values of the $X$ variable are assigned or selected, and then the values of the $Y$ variable are observed later?
        \begin{enumerate}
            \item Retrospective
            \item Prospective
            \item Introspective
            \item Cross-sectional
            \item Multi-sectional
        \end{enumerate} 
    \item What do you call a study in which the $X$ and $Y$ values are just observed, with no assignment or selection?
        \begin{enumerate}
            \item Retrospective
            \item Prospective
            \item Introspective
            \item Cross-sectional
            \item Multi-sectional
        \end{enumerate} 
    \item What do you call a study in which cases (there are $n$ cases) are selected based on their $Y$ values, and then $X$ is observed? 
        \begin{enumerate}
            \item Retrospective
            \item Prospective
            \item Introspective
            \item Cross-sectional
            \item Multi-sectional
        \end{enumerate} 

 \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item For a prospective design, which marginal totals are fixed by the design? Assume rows correspond to $X$ and columns correspond to $Y$.
        \begin{enumerate}
            \item Row totals
            \item Column totals
            \item Both row and column totals
            \item Neither row nor column totals
        \end{enumerate} 

    \item For a retrospective design, which marginal totals are fixed by the design? Assume rows correspond to $X$ and columns correspond to $Y$.
        \begin{enumerate}
            \item Row totals
            \item Column totals
            \item Both row and column totals
            \item Neither row nor column totals
        \end{enumerate} 

    \item For a cross-sectional design, which marginal totals are fixed by the design? Assume rows correspond to $X$ and columns correspond to $Y$.
        \begin{enumerate}
            \item Row totals
            \item Column totals
            \item Both row and column totals
            \item Neither row nor column totals
        \end{enumerate} 

% Now some questions from the textbook

% Oops, these are about odds and they have to wait.

%    \item In the text starting on p. 55, please do problems 2.3-2.7 in the text. 
%     \item Do 2.12 just parts $a$ and $b$.

    \item Do Problem 2.14 on p. 58 of the text. What homework problem from Assignment 3 shows you how to test the difference between those 2 percentages? What one number is missing?
    
    \item Do Problem 2.16, parts $a$ through $c$. Look at how many smokers there were in those days! Part $b$ is asking whether the study is prospective, retrospective or cross-sectional. 
        \begin{enumerate}
            \item \label{Rquestion} Carry out a likelihood ratio test for association of smoking and lung cancer. Use R, and do it the easiest way you can. Display the expected frequencies. Your printout should also include $G^2$, the degrees of freedom, and the $p$-value. Do you reject $H_0$ at $\alpha=0.05$? In plain, non-statistical language, what do you conclude? 
            \item Suppose that in this population the overall probability of lung cancer is one percent. Using this information and the data in the table, estimate the probability of lung cancer given that the person is a smoker.
        \end{enumerate}

    \item Do Problems 2.17, 2.18 and 2.21. For each question, state whether the study is prospective, retrospective or cross-sectional.

% Now one good question from me.

    \item For each of five Instagram models, a random sample of 200 followers is selected, and they are classified as Female, Male or Other.
        \begin{enumerate}
            \item What kind of design is this?
            \item What is $I$? What is $J$? % I=3 and J=5
            \item How many free parameters are there in the unrestricted model? % 10
            \item Make a 3 by 5 table and write probabilities (parameters or functions of the parameters) in the cells. Call the models A, B, C, D and E.
            \item We want to test whether the models attract different gender profiles. What is the null hypothesis?
            \item Does the number of equals signs in your null hypothesis equal $(I-1)(J-1)$?
            \item How many free parameters are there in the restricted model? % 2
            \item Write the restricted likelihood. Do not hesitate to use the notation for marginal totals.
            \item Based on your restricted likelihood, give the estimated expected number of male followers of Instagram model B. Is your answer on the formula sheet? 
        \end{enumerate} 



\end{enumerate} % End of all the questions

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\textbf{Please bring hard copy of your complete R input and output from Question \ref{Rquestion} to the quiz.}
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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistics, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\href{http://www.utstat.toronto.edu/~brunner/oldclass/312f22} {\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/312f22}}

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