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\begin{document}

{\Large Where are we?} \pz

We are in a position to do some ``real statistics.''  Everything we have
done up to the mid-term can be viewed as developing some
basic tools, language and point of view to build on. \pz

If you want, in one sentence, to know where we're going, it is
to understand topics like the ``margin of error'' for opinion polls.  \pb

More explicitly, so far almost all of the computations we have 
done have assumed that we completely understand
the distribution we are working with. \pz  Then we can compute
things like the mean, deviation, regression lines, sampling distributions,
etc.  \pz  But in the real world one hardly ever knows a complete data set
(except for small data sets such as exam scores).  \pz  The ``basic
practice of statistics''  (also known as {\em statistical inference}
is to start with some sample of a data set
and work towards getting a picture of that data set.  \pb

One important feature of this basic practice is that we will have to live
with uncertainty; \pz there is always a chance that even a well-taken
sample is ``unlucky''.  \pz

Within the context of this class, another feature will be that the material
will become more computational and example-driven, rather than
conceptual. \pb  

{\Large Confidence intervals} \pz

Confidence intervals are one of the most basic parts of the analysis
we do for the rest of the term. \pz

\begin{definition}
A confidence interval is a statement of the form ``There is a C\% chance
that the parameter we are trying to measure is between $X - D$ and
$X + D$.''  \pz  The quantity $D$ is called the margin of error. \pz
The value $X$ will most often be a statistical measure of the parameter
through some survey. \pb
\end{definition}

For example, ``There is a 95\% chance that the between
48 and 52\% of registered voters approve of the job President Bush  is
doing.'' \pz  This is pretty close to the standard statement ``50\% of registered
voters surveyed approve of the job President Bush is doing, with a margin
of error of $\pm 2\%$.  \pz  (The standard statement leaves out the ``95\%
chance bit'' - we will see that they are being a bit misleading.) \pb

We would like to construct confidence intervals just from having survey
data.  \pz But what we do as a first step is to construct them  {\em
assuming we already know the standard deviation of the distribution from 
which they are sampled} by using the Central Limit Theorem. \pz 

\begin{example}
Suppose we know that the standard deviation for new car purchases
is \$3000 \pz and we run a survey of 700 purchases which finds a 
mean purchase price of 23,142.  Find a confidence interval for the 
actual mean with a 95\% certainty.   \pz  or with 99.7\% certainty. \pz
\end{example}

Our method is to use the Central Limit Theorem to find the deviation
for the sample distribution; \pz then just note that if the sample
mean is within two deviations of the true mean then the true mean
is within two deviations of the sample mean (duh) and this happens
95\% of the time; \pz so we the confidence interval will 
be within two (adjusted) deviations of the survey mean. \pz

\begin{example}
We know that the standard deviation for heights of women over the
entire U.S.  cannot be more than $5$ inches.  Suppose
that we find a random sample of 400 women which has
an average height of $63.5$ inches.   Establish a confidence
interval for the true average height of women, with a confidence
of 90\%. \pz
\end{example}

It is remarkable how such a small sample can give a pretty small
margin of error with 90\% certainty. \pz

Another way to think about confidence intervals is to think about 
running multiple samples and keep track of the number of times 
that the mean is within our confidence interval.  We look at the
previous example. \pb




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