How can we assess the validity of a linear extrapolation? Is there a standard way of measuring the soundness of an extrapolation, like "statistical significance"?
The only way I can think of is to take the ratio of the sample with the population. For instance, if sample = 24 and population = 200 billion, their ratio is 0.00000000012 or 1.2e-10. How do we know if this is a valid linear extrapolation? Here, 24 is the number of galaxies Edwin Hubble observed in 1929 and 200 billion is the accepted number of galaxies in the observable universe.
What I’m trying to understand is this:
If Hubble observed just one galaxy and extrapolated that to the entire universe, no one would have believed him. If we assume that there are 200 billion observable galaxies then the ratio, sample/population is 1/2e11 = 0.0000000000050 or 5.0e-12.
If Hubble measured 2 galaxies, I believe, people would still question that extrapolation. Even instinctively, projecting a sample of 2 to 200 billion appears invalid.
If Hubble observed 10 galaxies, the ratio is 10/2e11 = 0.0000000000500 or 5.0e-11. That’s one order of magnitude difference from 1 galaxy, right?
In reality, Hubble observed 24 galaxies and extrapolated that linearly to 200 billion galaxies and no one objected to it. The ratio for 24 galaxies is: 24/2e011 = 0.0000000001200 or 1.2e-10. To me this is no different than observing only one galaxy and projecting it to 200 billion.
So, how do we determine that a linear extrapolation is valid? Is there a measure like “Statistical significance” for extrapolations?
For instance, in this site they write, regarding statistical significance, that,
if you’re taking a poll on how people plan to vote in an election, the accepted convention is that two standard deviations above or below the average, which gives a 95 percent confidence level, is reasonable. That two-sigma interval is what pollsters mean when they state the “margin of sampling error,” such as 3 percent, in their findings.
How do I calculate 2 sigma for this case?
Please note that, I never studied statistics (sorry if my terminology is off) but I would like to know the mathematics involved here. Thanks.