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\begin{document}

{\Large Notes on applying z-test for proportions, and more examples}  \pz


To do inference, we need to know we are reasonably close to a normal
distribution.  Here are some conditions:   \pz

\begin{itemize}
\item Our sample is a SRS.   This is hard, and is
usually the biggest reason to be skeptical about polls. \pz
\item The population is at least 10 times the sample size.  \pz
\item The sample size is ``large enough.''  (At least 15 successes and
15 failures.)   \pz
\end{itemize}

Unfortunately, even for relatively large $n$, this can be not so close
to Normal.    The way to fix this (recommended to always use):  the ``Plus four''
confidence interval.  \pz
Let $\overline{p} = \frac{\mbox{YESes} + 2}{n+4}.$  \pz

Then the C\% confidence interval is  between
$\overline{p} - z^{*} \sqrt{\frac{\overline{p} (1-\overline{p})}{n+4}}$ and
$\overline{p} + z^{*} \sqrt{\frac{\overline{p} (1-\overline{p})}{n+4}}$.  \pz
Hypothesis testing changes similarly.  \pz  We use the plus
four method in the next examples. \pb


\begin{example}
Suppose we take a SRS of 90 people under the age of 22, and discover 11 of them
have at some point had an STI.   Use this to estimate the proportion of the total
population which have had an STI?   \pz  (Note: half of the top ten most frequently   
reported diseases in the U.S., accounting for over four-fifths of all cases, are STI's).  \pz
\end{example}

\begin{example}
In a certain population, we expect 46\% of the
population to get a cold in a 3 month period.  \pz We give 264 volunteers
1000 mgs of vitamin C per day for 3 months.  At the end of the period
119 people (45\%) have gotten colds.  So the proportion of our sample
that have gotten colds is 
$\hat{p} = .45$.  \pz

Let $p$ be the proportion of people being treated with vitamin C
who will get colds.  We would like to know if there is evidence that
$p$ is smaller than $.46$. \pz
\end{example}  


\begin{example}
Establish some confidence intervals (both the usual and plus four) and
critique the methodologies
for polls found at:\\ 
{\tt http://www.usatoday.com/news/health/2006-05-29-marriage-poll\_x.htm}\\
{\tt http://www.usatoday.com/news/polls/tables/live/2006-05-01-poll.htm}\\
{\tt http://www.usatoday.com/news/world/2006-05-21-travelerspoll\_x.htm}
 \pb
\end{example}

\begin{example}
Find some polls on the web which publish their sample size and margin of error,
and determine with what certainty the number being measured is within
that margin or error.  \pb
\end{example}

\end{document}
 