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\begin{center}   
{\Large \textbf{STA 2101/442 Assignment 2}}\footnote{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/appliedf17} {\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/appliedf17}}}
\vspace{1 mm}
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\noindent
Except for Questions~\ref{satR} and \ref{mathR}, the questions on this assignment are practice for the quiz on Friday September 21st, and are not to be handed in. 

\begin{enumerate} 

\item \label{sat} In the United States, admission to university is based partly on high school marks and recommendations, and partly on applicants' performance on a standardized multiple choice test called the Scholastic Aptitude Test (SAT). The SAT has two sub-tests, Verbal and Math. A university administrator selected a random sample of 200 applicants, and recorded the Verbal SAT, the Math SAT and first-year university Grade Point Average (GPA) for each student. 
Because it is familia, convenient and not too unreasonable, we adopt the model $\mathbf{y}= X\boldsymbol{\beta}+\boldsymbol{\epsilon}$, where
\begin{itemize}
    \item $X$ is an $n \times p$ matrix of known constants with $n>p$ and the columns of $X$ linearly independent.
    \item $\boldsymbol{\beta}$ is a $p \times 1$ vector of unknown constants.
    \item $\boldsymbol{\epsilon}$ is an $n \times 1$ random vector with $E(\boldsymbol{\epsilon}) = \mathbf{0}$ and $cov(\boldsymbol{\epsilon}) = \sigma^2 I_n$.
    \item $\sigma^2>0$ is an unknown constant. 
\end{itemize} 
The least-squares estimate of $\boldsymbol{\beta}$ is $\widehat{\boldsymbol{\beta}} = (X^\top X)^{-1} 
X^\top \mathbf{y}$, the vector of predicted $y$ values is $\widehat{\mathbf{y}} = X\widehat{\boldsymbol{\beta}}$, and the vector of residuals is $\mathbf{e} = \mathbf{y} - \widehat{\mathbf{y}}$.
Give the dimensions (number of rows and number of columns) of the following matrices, specifically for the  SAT data. The answers are all numbers. 
    \begin{enumerate}
        \item  $\mathbf{y}$
        \item  $\boldsymbol{\beta}$
        \item  $X\boldsymbol{\beta}$
        \item  $(X^\top X)^{-1}$
        \item  $\widehat{\boldsymbol{\beta}}$
        \item  $\widehat{\mathbf{y}}$
        \item  $\mathbf{e}$
        \item  $\mathbf{e}^\top\mathbf{e}$
        \item  $\boldsymbol{\epsilon}\boldsymbol{\epsilon}^\top$
        \item  $X^\top \mathbf{e}$
    \end{enumerate}
 
 \item \label{perpendicular} For the general model of Question~\ref{sat} (not just the specific SAT example) show $X^\top \mathbf{e} = \mathbf{0}$. 
 
 \pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item Why does $X^\top\mathbf{e}=\mathbf{0}$ tell you that if a regression model has an intercept, the residuals must add up to zero?

 \item For the SAT data of Question~\ref{sat}, what null hypothesis would you test to answer the following questions? Give the answers in symbols.
     \begin{enumerate}
        \item Controlling for Math score, is Verbal score related to first-year grade point average?
        \item Allowing for Verbal score, is Math score related to first-year grade point average?
        \item Is either Verbal score or Math score (or both) related to first-year grade point average?
        \item Does expected GPA increase faster as a function of the Verbal SAT, or the Math SAT?
    \end{enumerate}

\item It is well known that people who graduate from university have higher lifetime earnings on average than those who do not. Mention at least one confounding variable that could have produced this result.

\item Suppose that volunteer patients undergoing elective surgery at a large hospital are randomly assigned to one of three different pain killing drugs, and one week after surgery they rate the amount of pain they have experienced on a scale from zero (no pain) to 100 (extreme pain).
    \begin{enumerate}
        \item What is the explanatory variable? There is just one.
        \item What is the response variable?
        \item Is this an experimental study, or observational?
        \item Why is it important for the patients to be unaware of which drug they are receiving? Relate this to the idea of a confounding variable.
        \item Is it also important for the physicians to remain unaware of what drugs their patients are getting? Why or why not?
    \end{enumerate}

\item In a large telecommunications firm, sales representatives working at kiosks in shopping malls get bonuses for signing up more customers to more expensive cell phone plans. In addition to financial incentives, the company decides to require additional training for the sales reps who performed worst last quarter. To assess the effectiveness of the training, they will carry out a statistical test (the mysterious matched $t$-test, in fact) to see whether the average performance of this group increases.
    \begin{enumerate}
        \item Your first thought is ``No! Don't do it this way!" Why? Plain language is not requested here. Just let me know that you understand the issue. Keep it short and sweet.
        \item How would you recommend that they assess the effectiveness of the training program? 
    \end{enumerate}

 \pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 \item High School History classes from across Ontario are randomly assigned to either a discovery-oriented or a memory-oriented curriculum in Canadian history. At the end of the year, the students are given a standardized test and the median score of each class is recorded. Please consider a regression model with these variables:
    \begin{itemize}
        \item[$X_1$] Equals 1 if the class uses the discovery-oriented curriculum, and equals 0 if the class uses the memory-oriented curriculum.
        \item[$X_2$] Average parents' education for the classroom.
        \item[$X_3$] Average family income for the classroom.
        \item[$X_4$] Number of university History courses taken by the teacher.
        \item[$X_5$] Teacher's final cumulative university grade point average.
        \item[$Y$] Class median score on the standardized history test.

    \end{itemize}

The full regression model (as opposed to the reduced models for various null hypotheses) implies 
\begin{displaymath}
    E[Y|X] = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \beta_4X_4 + \beta_5X_5.
\end{displaymath}
For each question below, please give 
    \begin{itemize}
        \item The null hypothesis in terms of $\beta$ values.
        \item $E[Y|X]$ for the reduced model you would use to answer the question. Don't re-number the variables.
    \end{itemize}

\vspace{2mm}

    \begin{enumerate}
        \item If you allow for parents' education and income and for teacher's university background, does curriculum type affect test scores? (And why is it okay to use the word "affect?")
        \item Controlling for parents' education and income and for curriculum type, is teacher's university background (two variables) related to their students' test performance?
        \item Correcting for teacher's university background and for curriculum type, are parents' education and family income (considered simultaneously) related to students' test performance?
        \item Taking curriculum type, teacher's university background and parents' education into consideration, is parents' income related to students' test performance?
        \item Here is one final question. Assuming that $X_1, \ldots, X_5$ are random variables (and I hope you agree that they are), 
            \begin{enumerate}
                \item Would you expect $X_1$ ro be related to the other explanatory variables?
                \item Would you expect the other explanatory variables to be related to each other?
            \end{enumerate}
    \end{enumerate}

 \pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\item \label{satR} The STA data described in Question~\ref{sat} are available 
\href{http://www.utstat.toronto.edu/~brunner/data/legal/openSAT.data.txt}
     {here}.
We seek to predict GPA from the two test scores. Throughout, please use the usual $\alpha=0.05$ significance level.
\begin{enumerate}
     \item First, fit a model using just the Math score as a predictor. ``Fit" means estimate the model parameters.  Does there appear to be a relationship between Math score and grade point average? 
    \begin{enumerate}
        \item Answer Yes or No.
        \item Fill in the blank. Students who did better on the Math test tended to have \underline{~~~~~~~~~~~} first-year grade point average.
        \item Do you reject $H_0: \beta_1=0$?
        \item Are the results statistically significant? Answer Yes or No.
        \item What is the $p$-value? The answer can be found in \emph{two} places on your printout.
        \item What proportion of the variation in first-year grade point average is explained by score on the SAT Math test? The answer is a number from your printout.
        \item Give a predicted first-year grade point average for a student who got 700 on the Math SAT. The answer is a number you could get with a calculator from your printout.
    \end{enumerate}
        
     \item Now fit a model with both the Math and Verbal sub-tests. 
    \begin{enumerate}
        \item Give the test statistic, the degrees of freedom and the $p$-value for each of the following null hypotheses. The answers are numbers from your printout.
            \begin{enumerate}
                \item $H_0: \beta_1=\beta_2=0$
                \item $H_0: \beta_1=0$
                \item $H_0: \beta_2=0$
                \item $H_0: \beta_0=0$
            \end{enumerate}
        \item Controlling for Math score, is Verbal score related to first-year grade point average?
            \begin{enumerate}
                \item Give the value of the test statistic. The answer is a number from your printout.
                \item Give the $p$-value. The answer is a number from your printout.
                \item Do you reject the null hypothesis?
                \item Are the results statistically significant? Answer Yes or No. 
                \item In plain, non-statistical language, what do you conclude? The answer is something about test scores and grade point average.
            \end{enumerate}
        \item Allowing for Verbal score, is Math score related to first-year grade point average?
            \begin{enumerate}
                \item Give the value of the test statistic. The answer is a number from your printout.
                \item Give the $p$-value. The answer is a number from your printout.
                \item Do you reject the null hypothesis?
                \item Are the results statistically significant? Answer Yes or No. 
                \item In plain, non-statistical language, what do you conclude? The answer is something about test scores and grade point average.
            \end{enumerate}
        \item Give a predicted first-year grade point average for a student who got 650 on the Verbal and 700 on the Math SAT.
        \item Let's do one more test. We want to know whether expected GPA increases faster as a function of the Verbal SAT, or the Math SAT. That is, we want to  compare the regression coefficients, testing $H_0: \beta_1=\beta_2$. 
        \begin{enumerate}
            \item Express the null hypothesis in matrix form as $\mathbf{L}\boldsymbol{\beta} = \mathbf{h}$. 
            \item Carry out an $F$ test.
            \item State your conclusion in plain, non-technical language. It's something about first-year grade point average. % Can't conclude that expected GPA increases at different rates as a function of Verbal SAT and Math SAT.
        \end{enumerate}
\end{enumerate}
\textbf{Please bring your printout to the quiz}.

\end{enumerate} % End of sat R questions

\item \label{mathR} Before the beginning of the Fall term, students in a first-year Calculus class took a diagnostic test with two parts: Pre-calculus and Calculus. Data are in the file
\href{http://www.utstat.toronto.edu/~brunner/data/legal/math1.data.txt} {\texttt{math1.data.txt}}. The variables are
\begin{itemize}
     \item Identification code
     \item Course: 1=Catch-up 2=Mainstream 3=Elite 4=NoResponse
     \item Score on pre-calculus part of diagnostic test
     \item Score on calculus part of diagnostic test
     \item High School GPA
     \item High School Calculus mark
     \item High School English mark
     \item University Calculus mark
     \item First language
     \item Sex
\end{itemize}
These are real data, with 776 missing values coded as NA. There are other rough edges, too; these have deliberately not been fixed. I'm not even sure if I remember what they all are, which makes this more interesting.

Your job is simple. Read the data and produce frequency distributions for the categorical variables and a table of means, standard deviations and sample sizes for the quantitative variables. But first, and as you do this, make a list of the strange things you notice about the data. Decide which of them should be fixed and which should be left alone. Fix the ones that you think should be fixed. Fix them in the way that makes most sense to you, so that the simple descriptive statistics you produce will be as meaningful as possible. All other things being equal, basing statistics on more data is better. 

It is okay to discuss this with classmates. In fact, we will all discuss this, identify the problems, consider the solutions, and then I will decide what to do about each problem and let you know. As you will see, some of these decisions will be semi-arbitrary, but at least that way we should get the same numerical answers when we analyze the data --- assuming we do it the same way.

\textbf{Please bring your printout to the quiz}.


\end{enumerate}


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Random explanatory variable question(s) next time.

\item 
    \begin{enumerate}
        \item 
        \item 
    \end{enumerate}