Posted by
statisticsdoc on
Dec 19, 2006; 9:41pm
URL: http://spssx-discussion.165.s1.nabble.com/Sample-Means-tp1072828.html
Stephen Brand
www.statisticsdoc.com
Richard,
Thanks for citing me on both sides of this discussion :) Let me say a little more about why I would accept that 1,000 cases can constitute a population, and under what conditions.
It is not too hard to imagine population definitions that encompass small numbers of people (e.g., all of the left-handed residents of the town of Exeter, Rhode Island; the Fall 2006 intake of a small college).
The question of whether you accept that 1,000 cases make up a population depends on the definition of the population. If these 1,000 cases are all of the potential members of the population, then the mean of those cases constitutes the population mean. Whatever random processes might have influence the mean score of that population, that score is the population parameter. We are not trying to estimate a parameter of a wider population from which we have obtained the 1,000 cases. In this instance, the one-sample procedure is justified.
Granted, you might say that the left-handed residents of Exeter, or the 2006 intake of a small college, constitute a sub-set of your population of interest, but then I think that you have to allow that these cases do not exhaust the potential membership of the population (which might constitute the left-handed population of Rhode Island, or the various cohorts of potential incoming first year students), and then your means become sample statistics, not population parameters. BTW, in this instance, the Exeter sample is not a very random one :)
It all depends on where the boundaries of the population are drawn.
Best,
Stephen Brand
Stephen Brand
-----Original Message-----
From: SPSSX(r) Discussion [mailto:
[hidden email]] On Behalf Of Richard Ristow
Sent: Tuesday, December 19, 2006 2:59 PM
To:
[hidden email]
Subject: Re: Significant difference - Means
To weigh in with two comments:
At 03:54 AM 12/19/2006, Spousta Jan wrote:
>The error of that 3.5 is about sqrt(1/1000) = 0,03 while the error of
>2.9 for students is about sqrt(1/500) = 0.045. That is both errors are
>of the same order of magnitude and the population error cannot be
>neglected in this case.
I'd like to second, and emphasize, this. Jan is clearly right here,
where the two groups are the same size. However, the same thing holds
when the sizes are quite different.
First, the t-test algorithm correctly allows for the increased
precision in measuring the mean in the larger group. Replacing it by a
constant only 'gains' you a little precision you don't really have.
Second, inequality of group size matters less than one might think.
Roughly, precision goes as the square root of sample size. (Under
'nice' conditions, that's exact: standard error of estimate goes as the
square root of sample size.) That means increasing the sample size
ten-fold leaves the SEE still 1/3 of the size it had - quite a long way
from letting it be considered a constant.
And at 10:42 AM 12/19/2006, Statisticsdoc (Stephen Brand) wrote:
>If your population consists of the 1000 students, then the mean of 3.5
>is a population parameter, and you would be justified is using the
>one-sample t-test suggested by John.
(This won't be quite fair to Stephen Brand, who'd also written
"Formally one should test the null hypothesis that the two samples have
the same mean, by using the independent groups t-test.")
There's a philosophical position, which I agree with, that will hardly
ever accept something like "[my] population consists of the 1000
students." The argument is that, even if those 1,000 students are all
you've ever seen or ever will see, their observed values constitute a
set generated by an underlying random mechanism, and that randomness
must be allowed for in estimation exactly as if you were aware of
100,000 similar students.
('Generated by an underlying random mechanism' is sometimes expressed
as 'drawn from a conceptually infinite population.' However, while this
is technically accurate, I don't blame anyone who considers a
'conceptually infinite population' a very odd notion.)
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