I'm sure some other stats expert out there will give a better
explanation, but here's my take on your questions:
[EMAIL PROTECTED] wrote:
>
> Do the null and alternative hypotheses have to be complements? In
> other words, can one set up a problem like:
>
> H0: d = 0
> H1: d <= 0
No, this is not a valid alternative hypothesis. By stating d <= 0 as
your alternative hypothesis, you are essentially saying that the
alternative hypothesis is the same as the null hypothesis, which is
bogus. The alternative hypothesis must either be d < 0, or d > 0, or d
!= 0. But strictly speaking, your null and alternative hypotheses do
not have to be complements. As long as your alternative hypothesis is
not contained in the set defined by the null hypothesis, the hypotheses
are valid.
Bear in mind also that you are not trying to prove the null hypothesis.
You never do that. You are trying to prove the alternative hypothesis
to see if the evidence is strong enough or not to reject the null
hypothesis.
That is why it is valid to have
H_0: d > 0
H_a: d < 0
or
H_0: d = 0
H_a: d < 0
For the above two hypotheses, you have the same rejection region. We do
not really care about the H_0 being larger or equal to 0. All we need
is the relationship between the alternative hypothesis and the null
hypothesis.
<clip>
> If you can go with the one-tailed test, does choosing it imply a
> different Type II loss function than in the two-tailed test?
I am not familiar with the term "loss function". Sorry but I do not
know what it means.
Assuming that it means "error", as in Type II error, the definition of
Type II error is the same for both one-tailed and two-tailed tests.
That is, the error of not rejecting H_0 when H_0 is false. The idea is
basically the same for the calculation of Type II error probability
[beta(mu')] for both one-tailed and two-tailed tests. Of course the
formula will be slightly different because when you look at a two-tailed
test, the Type I (alpha) error is on both sides of the tails in a
bell-shaped curve. Whereas in a one-tailed test, the probability of a
Type I error is just the area within one tail of the curve.
- Khai
> Are there
> any other hidden assumptions underlying the choice of a one-tailed test
> in this context?
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