On a quiz, I set the following problem to my statistics class:
"The manufacturer of a patent medicine claims that it is 90%
effective(*) in relieving an allergy for a period of 8 hours. In a
sample of 200 people who had the allergy, the medicine provided
relief for 170 people. Determine whether the manufacturer's claim
was legitimate, to the 0.01 significance level."
(The problem was adapted from Spiegel and Stevens, /Schaum's
Outline: Statistics/, problem 10.6.)
I believe a one-tailed test, not a two-tailed test, is appropriate.
It would be silly to test for "effectiveness differs from 90%" since
no one would object if the medicine helps more than 90% of
patients.)
Framing the alternative hypothesis as "the manufacturer's claim is
not legitimate" gives
Ho: p >= .9; Ha: p < .9; p-value = .0092
on a one-tailed t-test. Therefore we reject Ho and conclude that the
drug is less than 90% effective.
But -- and in retrospect I should have seen it coming -- some
students framed the hypotheses so that the alternative hypothesis
was "the drug is effective as claimed." They had
Ho: p <= .9; Ha: p > .9; p-value = .9908.
Now as I understand things it is not formally legitimate to accept
the null hypothesis: we can only either reject it (and accept Ha) or
fail to reject it (and draw no conclusion). What I would tell my
class is this: the best we can say is that p = .9908 is a very
strong statement that rejecting the null hypothesis would be a Type
I error. But I'm not completely easy in my mind about that, when
simply reversing the hypotheses gives p = .0092 and lets us conclude
that the drug is not 90% effective.
There seems to be a paradox: The very same data lead either to the
conclusion "the drug is not effective as claimed" or to no
conclusion. I could certainly tell my class: "if it makes sense in
the particular situation, reverse the hypotheses and recompute the
p-value." Am I being over-formal here, or am I being horribly stupid
and missing some reason why it _would_ be legitimate to draw a
conclusion from p=.9908?
--
Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://oakroadsystems.com
My reply address is correct as is. The courtesy of providing a correct
reply address is more important to me than time spent deleting spam.
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