So what good are these transformed statistics? As we said, we know what they should be close to if our statistic is close to the true parameter. The miracle is that (if certain assumptions are met) statisticians have determined mathematically intervals of the real line that a transformed statistic will fall into with specified probability.
For example, the first transformed statistic is labeled Z because statisticians have shown that if the population is normally distributed, then the transformed statistic has the Z distribution (the standard normal curve). Thus if we repeatedly selected random samples of size n and and calculated Z for each one, then we know that 95% of the samples will have a Z between -1.96 and 1.96. (You will use the ``Sampling Distribution'' concept lab to experiment with this idea).
Thus, how close is 
to 
in this situation? We saw earlier
in this week that 95% of the area under a Z curve fals between
-1.96 and 1.96. This tells us that
95% of all samples will have
that is, 95% of all samples will have 
within
. For example, 95% of all samples
of 25 IQ's (remember that IQ's are thought to be normally
distributed with 
and 
) will have
in 
, that is, 95% of all samples will have
within 
3 of .