Download 1 One-sided hypothesis tests on means of normal distributions For

One-sided hypothesis tests on means of normal distributions
For these tests the "detection probability" is an extension of the more familiar concept
of "statistical power"—the frequency with which a test will reject a false hypothesis
over the region where the hypothesis is actually false. However, detection probability
is defined over the whole range of possible differences of means (as measured by the
“effect size”), regardless of whether the hypothesis is true or false.
Let’s first consider the case where there is a single group of data. There are two
complementary sets of hypotheses that can be tested. In the first set the hypothesis to
be tested postulates that the true mean (µ) is less than the standard value (µ0). This is
known as the "inferiority hypothesis" (H: µ ≤ µ0). In this case the sample mean will
have to be some way above µ0 before the hypothesis is rejected, i.e., a difference is
"detected". This is because rejection of the inferiority hypothesis will only occur if
results are "beyond reasonable doubt". Accordingly, the detection probability is high
when the true effect size is positive and large, and small if the effect size is large and
negative (reflecting the fact we would seldom obtain a sample mean rather larger than
µ0 when the population mean is much lower than that value). The converse is true for
testing the complementary "superiority hypothesis" (H: µ ≥ µ0), postulating that the
true mean (µ) is greater than the standard value (µ0). In this case the sample mean
will have to be some way below µ0 before the hypothesis is rejected—again because
rejection of the hypothesis will only occur if results are "beyond reasonable doubt".
Just how much the sample mean must be above or below µ0 in order for rejection to
occur depends on the sample standard deviation and on the number of samples—tests
tend to be more powerful with larger sample sizes. Note that if either hypothesis is not
rejected, it can be accepted (but two-sided point null hypotheses cannot logically be
accepted).
For the case where there are two independent groups of samples (x and y), we replace
µ in the discussion above by the difference in the group means, denoted as δ = µx – µy.
If the samples are paired, the test is performed on the differences between each pair,
so it is effectively a one-group test.
Full technical details, including theory and associated calculation methods, are given
in McBride (2005, Using Statistical Methods for Water Quality Management: Issues,
Problems and Solutions, Wiley, New York).
One-sided test explanation 18 November 05.doc
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