Indoor? wrote:
belial wrote:
I'm confident good research would support my thesis, and my confidence means a lot. A real lot of lots.
Well considering there aren't that many who run 10 or 10:10 it's a little odd.
Let's look at 2012.
No one even ran 10:00. Randomly we'll pick the 2nd girl. Laura Hollander, she ran 10:10.
Randomly we'll pick the 10th girl, who ran 10:21. Molly Seidel. You say there is a greater than 50% chance that Laura Hollander would end up faster than Molly Seidel.
Population A was constructed from female runners that have a PR that is around 10:00. Each time we randomly draw a runner from this population for a race, the runner will realize an uncertain race time denoted by the random variable X. X has a distribution of values such that a random race time X = x will be realized with some probability p(x).
Population B was constructed from female runners that have a PR around 10:10 and we denote the population's uncertain race times by the random variable Y.
What can we say about each of these populations as defined? Not much. We don't know the shape of either distribution of X and Y, e.g., we don't know Pr[X < Y] -- it could theoretically, given what we know so far, take on any value in the range (0,1). The only thing we can really say at this point is that the PRs are different, but each time we draw a value from either distribution (each time a runner races), a new PR may result. However, intuitively, we assume that, given populations A and B, a randomly chosen runner from A is expected to do better than a randomly chosen runner from B, i.e., Pr[X < Y] > 0.5. That is to say, we assume that the PR metric that we constructed the populations from is generally a good indicator of racing ability, even though, for instance, some of the runners in A just had a fluke PR.
Finally, since we assume the future will be like the past, we assume that in the future Pr[X' < Y'] > 0.5 also, where X' is the uncertain race times of future runners from population A and likewise for Y'.
Now, I've stated my assumptions: (1) the PR is a good indicator of racing ability and (2) the future is like the past. Both of these assumptions are loaded with ambiguity and uncertainty, e.g., the future is like the past may take the form "exceptional HS runners tend to do worse in the future than simply very good HS runners." However, all things else being equal (i.e., I have no did any research to tell me otherwise), my conclusions seem to follow from the stated assumptions.
A randomly selected girl from population A is expected to do better than a randomly selected girl from population B. Also, note that it doesn't matter whether any HS girls have ran these PRs since the populations were given by a definition, not by an actual sample.