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Trevor,

You are correct, Jan and I are discussing whether the normal
distribution or the t distribution apply.  The normal distribution is
used when you're working with a single observation, the t when you're
working with a mean of more than one.  We're working here with a mean
of six observations, and it makes sense that we can estimate this more
precisely with six observations than one, and that the precision of the
estimate depends on the number of observations.

Jan was using the test statistic for the normal distribution, usually
denoted z, with y equal to the value of a _single_ observation, u the
actual or parametric mean and s the standard deviation is:

z = (y - u) / s,

whereas I was using the test statistic for the t distribution, denoted
t, with Y equal to the observed mean of a set of observations, u their
parametric mean, s the standard deviation and n the number of
observations is:

t = (Y - u) / (s / the square root of n).

For the first statistic the error term is the standard deviation, for
the second the standard error of the mean.  I might note that implicit
in the z statistic is that n=1.

This is not a "paired t test."  It is not even a t test for separating
two means.

To me, the only point in question was whether the standard deviation
could be estimated from the mean of the observations, which it appears
Jan and I agree can be done, and, equivalently, that the variation
between a set of observations was the same size as the variation
within.

You can find a discussion of normal and t distributions in elementary
statistics texts.  As I indicated previously, the statistics are very
robust to departures from assumptions and, in my opinion, it's likely
that the data on which Jan is reporting conform fairly closely to the
expected distribution.  But I also agree with your points about
outliers.

Finally, perhaps I have managed to refute Jan's bleak picure, as he
wished, and shown that, in the hands of a competent observer, 95% of
the time a lunar will be accurate to within 25" of arc with 6 replicate
observations, and 99.9% of the time accurate to within 44" of arc.

Fred

On Jan 1, 2004, at 7:26 AM, Trevor J. Kenchington wrote:

> Fred & Jan,
>
> With apologies for butting into your exchange but you two seem to be
> misunderstanding one another:
>
> Fred: What you are suggesting is, I think, a paired-comparison t-test
> of
> the null hypothesis that the mean of all possible positions estimated
> by
> lunars is not different from the mean of all possible positions
> estimated by chronometer, using a sample of paired position estimates,
> one each by lunar and chronometer (where, of course, each "estimate" is
> actually based on multiple sextant observations on the same day). That
> would be one way to see whether there was a bias in the lunar estimates
> of position but it would not address Jan's interest in the average
> absolute magnitude of the differences between the lunar and chronometer
> estimates.
>
> For that, Jan already has someone else's estimate of the "probable
> error", meaning the absolute difference that will be exceeded by 50% of
> paired position estimates (on average). To get from that value to the
> standard deviation of the errors in the lunars, he has assumed that the
> chronometer positions are exact (which is likely close enough to the
> truth) and has consulted the percentage points of the _Normal_
> distribution, _not_ the t-distribution which you turned to. Armed with
> the standard deviation, he has again turned to the Normal distribution
> to read off the error that should occur in 3 estimates in every
> thousand
> (i.e. 1-0.997 = 0.003 of lunar estimates).
>
>
> The problem with that, as I have noted in earlier messages, is that the
> errors in the lunars are unlikely to be exactly Normally distributed.
> Otherwise, I think that Jan has followed the right path in trying to
> estimate the magnitude of the errors that a skilled navigator might get
> with lunars. He doesn't need hypothesis testing and hence he does not
> need the t or F distributions.
>
>
> Trevor Kenchington
>
>
> --
> Trevor J. Kenchington PhD                         Gadus@iStar.ca
> Gadus Associates,                                 Office(902) 889-9250
> R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
> Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555
>
>                     Science Serving the Fisheries
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>

   

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