> 3) When n is greater than 30 and we do not know sigma, we must estimate
> sigma using s so we really should be using t rather than z.
you are wrong. you use t-distribution not because you don't know sigma,
but because your statistic has EXACT t-distribution under certain
conditions. I know that the textbook says "if we knew sigma then the
distribution would be normal, but because we used s instead the
distribution turned out to be t". It does not say how exactly it becomes
t, so you make the conclusion: use t instead of normal whenever you use s
instead of sigma. But it's wrong, it does not go like this.
when you don't know underlying distribution of the sample you may use
normal distribution (under certain regularity conditions),
as an APPROXIMATION to the actual distribution of your statistic.
approximate distribution in most cases is not parameter-free, it may
depend, for example, on unknown sigma. in such situation you may replace
the
unknown parameter by its consistent estimator.the approximate
distribution is
still normal. think about it as iterated approximation. first you
approximate the actual distribution by N(0,sigma^2), then you approximate
it by N(0,S^2), where S^2 is a consistent estimator for sigma. there are
formal theorems that allow you to do this kind of thigs.
The essential difference between two approaches is that the first one
tries to derive the
EXACT disribution, second says I will use APPROXIMATION.
number 30 has no importance at all, throw away all the tables you have. I
cannot believe they still teach you this stuff. I wish it was that
simle:30!
Your confusion is the result of oversimplification and desire to provide
students with simple stratagies which present in basic statistics
textbooks. I guess it makes teaching very simple, but it mislead students.
Your confusion is an example. The problem is that there is no simple strategies,
and things are much-much more complicated than they appear in basic textbooks.
Basic text books don't tell you the whole story, and they don't even try,
because you simply cannot do this at their level. Don't make any strong
conclusions after reading only basic textbooks.
In practice, in business and economics statistics, nobody uses
t-tests, but normal and chi-square approximations are used a lot. The
assumptions that you have to make for t-test are too strong.
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