R. Webster West and R. Todd Ogden

University of South Carolina

Journal of Statistics Education v.6, n.3 (1998)

Copyright (c) 1998 by R. Webster West and R. Todd Ogden, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

This applet illustrates the fundamental principles of statistical
hypothesis testing through the simplest example: the test for the mean
of a single normal population, variance known (the *Z* test).

The basic set-up of the test is this: using only *n* independent
observations
*X*_{1}, *X*_{2},..., *X _{n}*
from a normal
distribution with unknown mean (but known variance), the task is to
decide whether or not to reject a null hypothesis
in favor of some alternative hypothesis. In most
applications, there are only three alternative hypotheses of interest:

and then rejecting the null hypothesis if the appropriate condition is satisfied. In the order of the alternative hypotheses above, the null hypothesis is rejected if

- ,
- ,
- ,

This hypothesis testing procedure is set up to give the null
hypothesis the "benefit of the doubt;" that is, to not reject the null
hypothesis unless there is strong evidence to support the alternative.
If *H*_{0} is true, the above test statistic follows a
standard normal distribution, so the probability of erroneously rejecting
*H*_{0} is just
.
If *H*_{a} is true, however, the test statistic
*Z* does not follow a standard normal distribution -- it
follows a normal
distribution with a different mean, and thus, the probability of
(correctly) rejecting the null hypothesis is larger than
.
This probability is knows as the power of the test, and it depends
on the true value of
.
(Clearly, a test would have more power for an extreme value of
than for a
that is very close to
.)

To use this applet, you must specify the hypothesized mean
, the true mean , and the value of
,
and select the
appropriate alternative hypothesis. Clicking on the **Show it!**
button will give a plot. The black curve represents the
distribution of the test statistic when the null hypothesis is true.
The portion shaded in red represents the probability of falling beyond
the cut-off point(s) when the null hypothesis is true (the Type I
error rate, or
). The blue curve represents the
distribution of the test statistic under the particular value of
you selected.
The blue shaded area represents the power of the test for that
particular value of
. Note that the region
shaded both blue and red appears purple.

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