On Mon, 8 May 2000, Khai L. Lai wrote:
> 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.
Correct so far. (I assume by "!=" you mean "not equal".)
> But strictly speaking, your null and alternative hypotheses do
> not have to be complements.
Strictly speaking, the set of hypotheses
(null plus one or more alternatives) must be mutually exclusive and
exhaustive, which I suspect is what you mean by "complements". Otherwise
rejecting the null hypothesis does not afford a logical reason for
accepting the alternative.
> As long as your alternative hypothesis is not contained in the set
> defined by the null hypothesis,
Necessary but not sufficient.
> the hypotheses are valid.
Hypotheses, as unconfirmed statements
about the universe of discourse, are ALWAYS valid. What you presumably
meant here was the _set_ of hypotheses comprising null & alternative.
> Bear in mind also that you are not trying to prove the null hypothesis.
> You never do that.
I think your heart is in the right place, even
if your language isn't, quite: that is, I suspect you intend "prove" to
have its corrupted meaning "demonstrate", rather than its root meaning
"to test". One is, in fact, testing the null hypothesis: that's the
only hypothesis for which a sampling distribution can be constructed,
because it's the only hypothesis in the set that specifies a parameter
value (that is, it's the only one with an "=" sign). To put it in other
terms, one is proving whether the null hypothesis be credible, on the
basis of the evidence at hand.
> You are trying [ to prove the alternative hypothesis ]
> to see if the evidence is strong enough or not to reject the null
> hypothesis.
Omitting the infinitive phrase I've enclosed [in
brackets], this statement is correct.
> That is why it is valid to have
> H_0: d > 0
> H_a: d < 0
No, it is not. No "=" sign in H_0.
> or
> H_0: d = 0
> H_a: d < 0
This is a not uncommon formulation, but is technically
incorrect since the possibility d > 0 is not represented in the set of
hypotheses. If the alternative is to be one-sided ( d < 0 in this
example), the null must also be one-sided (here, d >= 0 ).
> For the above two hypotheses, you have the same rejection region.
^^^^^^^^^^
You presumably meant "sets of hypotheses". But since the first set is
not a valid _set_ of hypotheses (for a formal hypothesis test), this
assertion is more or less meaningless, if not incorrect.
> We do not really care about the H_0 being larger or equal to 0.
Which implies that we need some decision rule for what to conclude should
the data imply that d > 0. "We do not really care about d being
larger or equal to 0" (as you should have phrased it!) implies that
d > 0 has the same meaning and effect as d = 0 ; which in turn implies
that both assertions should be included in the statement of H_0.
> All we need is the relationship between the alternative hypothesis and
> the null hypothesis.
... Once one has decided on a suitable set of
null and alternative hypotheses.
< snip, the rest >
-- DFB.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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