I don't believe anyone has bothered to define what they mean by a z-test.
There are two issues that must be dealt with: (1), What statistic is to be
used
and (2), what distribution is to be used to assess the size of that statistic.
I contend that a z "statistic," viz., (Ybar-mu0)/(sigma/sqrt(n)), is pretty
useless
since it almost never is a "statistic," i.e., it involves an unknown
parameter, sigma.
So I spend maybe 2 minutes of class time on that "statistic" and go on to
(Ybar-mu0)/(s/sqrt(n)) which I would call a t statistic. I then would say if
n is large we may use the normal distribution to assess the size of this
statistic
(using an _extension_ of the Central Limit Effect).
Finally, I get to the t distribution for assessing the size of the t statistic
when dealing with smaller sample sizes (and under certain additional
assumptions).
Jon Cryer
At 08:46 AM 4/20/01 +1000, you wrote:
>There is at least two very good pedagogical reasons for teaching z
>tests. Both the z and t tests are based on normality - the t test is
>used only because the model standard deviation is unknown or rather,
>there is no assumed value for it. Whether or not this is in practice
>'always' the case is irrelevant from the point of view of understanding
>what is going on. 'In an ideal world' we would do z tests! But in
>practice we usually cannot assume a value for sigma, so we are forced to
>use a t test. This is less powerful than the z test. So we pay a price
>for the lack of knowledge - if we don't know sigma, we pay for this in
>lowered power of the test.
>
>As a general principle, this is a fundamental aspect of statistics - I
>rate it as one of the reasons why students learn statistics! Lack of
>knowledge costs!
>
>So the two good reasons are - that the z test is the basis for the t,
>and the understanding that knowledge has a very direct value.
>
>I hasten to add that 'knowledge' here is always understood to be
>'assumed knowledge' - as it always is in statistics.
>
>My eight cents worth.....
>
>Alan
>
>
>--
>Alan McLean ([EMAIL PROTECTED])
>Department of Econometrics and Business Statistics
>Monash University, Caulfield Campus, Melbourne
>Tel: +61 03 9903 2102 Fax: +61 03 9903 2007
>
>
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Dept. of Statistics www.stat.uiowa.edu/~jcryer \ \_University
and Actuarial Science office 319-335-0819 \ * \of Iowa
The University of Iowa dept. 319-335-0706 \ /Hawkeyes
Iowa City, IA 52242 FAX 319-335-3017 |__________ )
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"It ain't so much the things we don't know that get us into trouble.
It's the things we do know that just ain't so." --Artemus Ward
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