Stan Brown wrote:
> 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.
I don't understand where they get the .9908 from. Whether you test a
one-or a two-sided alternative, the test statistic is the same. So the
p-value for the two-sided version of the test should be simply twice
the p-value for the one-sided alternative, 0.0184. Hence the paradox
you speak of is an illusion.
Unfortunately for you, the two versions of the test lead to different
conclusions. If the correct p-value is given, I would give full marks
(perhaps, depending on how much the problem is worth overall,
subtracting 1 out of 10 marks for the nonsensical form of Ha).
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