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\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Lab Day 1/Discussion -- Confidence Intervals}
\centerline{February 15, 2010}
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{\it Background}
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We have now introduced the ``large-sample'' confidence intervals
for means and differences of means, proportions and differences
of proportions.  We use the point estimators from Table 1 in
Section 8.3 to derive the confidence interval estimators.  

For example, the large-sample level 
$(1-\alpha)\times 100\%$ two-sided confidence interval for a mean 
is 
$$\mu = \overline{Y} \pm z_{\alpha/2} {\sigma\over \sqrt{n}}\leqno(1)$$
When we don't know the population $\sigma$, we
approximate with the sample standard deviation $s$.  

Nevertheless, the concept of confidence intervals is a notoriously 
``slippery'' one.  It is easy to go from the intuitively 
appealing ``confidence level'' $(1-\alpha)\times 100\%$
to statements that seem to be saying the same thing, but
are misleading at best, and completely meaningless at
worst.   To use this idea reliably to make inferences from
real-world data, it is very important to understand {\it exactly} 
what it means.
The purposes of today's lab/discussion are to 
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\item{1)} Reinforce the exact meaning of the confidence 
level $(1-\alpha)\times 100\%$ by sampling and analyzing 
the confidence intervals we get.  
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\item{2)} Look critically at some of the misleading
or meaningless ways that the idea of a confidence interval
can be ``abused.''

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{\it Getting Started -- Relevant Maple}
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In Swords 136, log on the campus network (so you have access
to your network P: drive disk space, the printer, etc.).  
Open a browser window, go to our course homepage, click 
on the Maple package link, and save the file to your network 
P: drive, under the name {\tt MSP.map} (without the period:) .  

Now, open a Maple window, and read in the statistics package
using 
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\centerline{\tt read "p:MSP.map";}
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Our class package contains a procedure 
called {\tt CIPlot} which creates a graphic illustrating
the meaning of confidence intervals.  The procedure
uses the formula (1) above (substituting $\sigma = s$
with each sample) to construct confidence intervals for the
mean from samples from a normal population.  Consult the 
online documentation for the Maple package (link
on course homepage) to see how it works, and how
to interpret the output.  You can also examine the source
code in the {\tt MSP.map} file if you're interested.

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{\it Lab/Discussion Questions}
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(You will probably want to create a Maple worksheet containing
the requested plots, plus answers to the questions below.)

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\item{1)}  Use the {\tt CIPlot} procedure to generate the plot
showing 100 $95\%$ confidence intervals for the mean, 
generated from samples of size $N = 35$ from a normal
population with $\mu = 20$ and $\sigma = 4.5$. 
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\item{2)}  Explain
what your plot shows.  In particular, how many of the intervals
contain $\mu = 20$ and how many do not?  How does that relate
to the $95\%$ confidence level?  
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\item{3)}  Will the plot {\it always} show exactly 95 intervals
containing the population mean $\mu = 20$?  In what sense do 
$95\%$ of the confidence intervals contain $\mu = 20$.  (You
may want to generate several more plots using {\tt CIPlot} --
each call uses new random samples, so the results will be different.)
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\item{4)} (Try to answer this before going to Maple, then use the 
procedure to check your answer.)  Suppose you now use {\tt CIPlot}
to study the $90\%$ confidence intervals for the mean (i.e. new 
$\alpha$), 
generated from samples of the same size $N = 35$ from the same 
normal population with $\mu = 20$ and $\sigma = 4.5$.   
What will change?
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\item{5)}  (Try to answer this before going to Maple, then use the 
procedure to check your answer.)  Suppose you now use {\tt CIPlot}
to study the $95\%$ confidence intervals for the mean (same $\alpha$
as in 1 above), 
generated from samples of the new size $N = 70$ from the same 
normal population with $\mu = 20$ and $\sigma = 4.5$.   
What will change (apart from the fact that the computation will
take somewhat longer)?  When you check your intuition, 
be sure to look carefully at the horizontal axis scales.
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\item{6)}  Suppose you now use {\tt CIPlot}
to study the $95\%$ confidence intervals for the mean (same $\alpha$
as in 1 above), 
generated from samples of the new size $N = 10$ from the same 
normal population with $\mu = 20$ and $\sigma = 4.5$.  Run
the procedure 10 times and compute the total number of intervals
that contain $\mu = 20$.   (You only need to show the graph for
the last one in your lab report.)
The results here should be different from the other cases.
Why is that true?  (Recall that when we don't know $\sigma$, we approximate
it by the sample standard deviation $s$.  What is the exact
distribution of ${\overline{Y} - \mu \over s/\sqrt{N}}$?  What
do you conclude here?)
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\item{7)}  A common {\it misconception} about confidence
intervals can be stated as follows:  ``When you compute
the $95\%$ confidence interval for the mean from a particular 
sample, there's a $95\%$ chance that the population mean is 
contained in your interval.''   This statement (taken literally) is actually 
meaningless -- why?  What could you do to modify the statement 
so that it makes sense and is a true statement about confidence 
intervals?
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\item{8)}  Another common {\it misconception} about confidence
intervals can be stated as follows: ``When you increase the
sample size $N$ the width of the $(1-\alpha)\times 100\%$
confidence interval always decreases.''  This statement is actually 
false -- why? What could you do to modify the statement 
so that it makes sense and is a true statement about confidence 
intervals?
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\item{9)}  A final common (and tempting) {\it misconception} about
confidence intervals deals with this situation. Say we are using
the interval to decide whether evidence from samples supports the
hypothesis that a population mean $\mu$ has a particular value
$\mu_0$.  If the value $\mu_0$ is contained in a confidence
interval but is far from the midpoint (even very close to one
endpoint), it is tempting to think 
that the evidence indicates possibly $\mu\ne\mu_0$.  On Friday, 2/12
we said that this is an {\it incorrect} deduction in this situation.  
However, we did not give a complete reason.
Look carefully back at your results from part 1.  Explain why concluding
$\mu\ne\mu_0$ on the basis of one confidence interval where
$\mu_0$ is close to an endpoint is {\it not a correct conclusion}.
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{\it Assignment}
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Group writeups due in class Monday, February 22.
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