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Greg,
The standard error of the means for with and without sight tube are
+/- 0.143 and +/- 0.157 at the 95 percent level and the variance ratio, F =1.21.

Conclusion: no statistically significant difference between your two series.

and Frank, I've repeated a naked eye vs telescope series using a Tamaya from 
the early 40's. The horizon mirror is half glass and half fresh air, so there 
is no paint line to confuse the issue. Here are the results:

Naked eye:
Mean 2.817 min +/- 0.179  => range of 2.99 to 2.638 ie 90 percent probability 
that the mean lies in this range.
SD = 0.50 Variance = 0.25

x 4 Galilean 'scope
Mean 3.213 +/- 0.084 => range of 3.297 to 3.129
SD = 0.235  Variance = 0.055

Variance ratio = 4.54. Tabulated value at 2 % level 2.39

My conclusion is that the tighter limits for the telescope are real and that 
index error should be determined using the sighting aid that is to be used to 
make the rest of the observations. I think that strictly speaking we are not 
entitled to compare the means because the variances are so different, but it 
is fairly clear that the means _are_ different and Douglas has suggested a 
reason for it. Tentatively, I would suggest that the vernier acuity 
hypothesis is looking somewhat shaky.

Statistics is coming back to me, but I hope there's someone out there who is 
checking my figures and conclusions 'cos I'm not confident with maths.

Bill Morris
Pukenui
New Zealand



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