On 2003-08-29 05:40:37 +0100 Branden Robinson <branden@debian.org>
wrote:
> Here are the results of the survey.
>
> possible non-
> developers developers developers
> -----------------------------------------------------------------
> option 1 ("no") 18 3 22
> option 2 ("yes") 1 0 1
> option 3 ("sometimes") 8 4 4
> option 4 ("none of the above") 1 0 1
Here is the summary of your friendly local statistical analysis:
I conclude that there is a probability of less than 1 in 1000 that the
above total vote for option 1 would have been obtained by pure chance
if there was no majority for option 1 over all others. I believe that
common practice in matters of belief is to use a 10% level (ie, look
for a probability of greater than 1 in 10). I assumed that the
distribution is binomial and that the above is representative of
possible voters.
Technical details of the test:
H_0 : p = 0.5
H_1 : p \gt 0.5
This is a one-tailed test. We are assuming a binomial distribution
and have n=63 observations. $np=31.5 \gt 5$ and $np(1-p) = 15.75 \gt
5$, so we can approximate the binomial distribution with a normal
distribution. Because the variance of the distribution under the null
hypothesis is known, we perform a Z-test. At the 5% level, the
critical region for a one-taled Z-test is Z > 1.96. At the 0.1%
level, the region is Z > 3.291.
The test statistic for the Z-test is $Z = \frac{x - \mu}{\sqrt{\sigma
/ n}}$, where $x$ is our obtained vote for option 1, so this is $Z =
\frac{43 - 31.5}{\sqrt{15.75 / 63}} = \frac{11.5}{\sqrt{1/4}} = 23$.
Clearly, this is greater than 3.291 and I reject $H_0$ in favour of
$H_1$ on the basis of the evidence used.
Notes: this test cannot be used safely to test for unanimity (ie H_0:
p = 1) because it would violate assumptions for the normal
approximation to the binomial. I cannot find a useful test of that
for such small numbers of possible outcomes. My initial suggestion of
chi-squared would have tested for a relationship between
developer/non-developer and the option chosen, which might be
interesting, but wasn't asked for.
About the author: MJ Ray was awarded a Bachelor of Science degree in
Mathematics with first class honours from the University of East
Anglia in 1997, after studying the mathematics with statistics
programme. He currently works as a consultant and performs
statistical analysis as part of his work, but this is rather different
to that, is unchecked and might be buggy, so he offers absolutely no
warranty on it. He is a Debian developer and sometimes writes about
himself in the third person.
--
MJR/slef My Opinion Only and possibly not of any group I know.
http://mjr.towers.org.uk/ jabber://slef@jabber.at
Creative copyleft computing services via http://www.ttllp.co.uk/
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