, F
The basic idea of statistical inference is that we can determine
(using what is called sampling distributions) the likely values of a number that
measures how far a statistic is from the corresponding parameter. For
example, we can measure how far the statistic 
is from the parameter
by calculating
the number (called a ``transformed statistic'')
and noting that if is close to , then Z should be
close to 0. Similarly, we can measure how close 
is to 
by calculating
which should be close to n-1 if is close to (we will
see in a minute why we use the symbols Z and to represent the
numbers).
In the table below, we write down a number of transformed statistics
and what they should be close to. You may wonder why we use these
transformations rather than some simple measure of distance such as
. The answer is that statisticians have learned over the
past 100 years that the more complicated transformations listed in the
table allow them to find the desired likely values while simple
distance measures are much more difficult to work with.
