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\title{CSCI 145 Problem 1: The Correlation Floor}
\author{} % TODO: Put your name here
\date{\today}

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\subsection*{Submission Instructions}

Please upload your work to Gradescope by \textbf{11:59pm on DATE}. % TODO: date
\begin{itemize}
\item Turn in a single PDF of \textit{handwritten} solutions. Typed (including \LaTeX) solutions will not be accepted.
\item Problems are grouped into three-week \textbf{cycles}. Once per cycle, you will meet with the grader and present the solution to \emph{one} problem from that cycle, selected at random when you arrive --- so bring your work and be ready to explain any of them.
\end{itemize}

\newpage \section*{Problem 1: The Correlation Floor \hfill \textnormal{2 points}}

A poll asks $n=10{,}000$ people a yes/no question and reports
$\bar{X}=\frac{1}{n}\sum_{i=1}^n X_i$, where every response has variance
$\Var(X_i)=\sigma^2$.  The usual margin-of-error calculation treats the
responses as independent, but suppose instead that every distinct pair has
the same small correlation $\rho$, so
$\Cov(X_i,X_j)=\rho\sigma^2$ for $i\neq j$.

Give a mathematical audit of the claim that ``a large enough poll can make
sampling error arbitrarily small.''  Your audit should derive the exact
variance of $\bar{X}$ from covariance bilinearity, identify what remains as
$n\to\infty$, and quantify the effect for $\sigma=\frac12$: determine the
value of $\rho$ for which a poll of $10{,}000$ has standard deviation $0.03$.
Finally, compute the size $n_{\mathrm{ind}}$ of an independent poll having the
same variance as an equicorrelated poll with parameters $(n,\rho)$, and use
your formula to explain whether collecting more responses from the same
population can remove the three-point uncertainty.

\paragraph{Results you may use.}
Covariance is bilinear, $\Var(Z)=\Cov(Z,Z)$,
$\Var(aZ)=a^2\Var(Z)$, and correlation is covariance divided by the two
standard deviations.

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