harry wrote:Suppose that there is a theory that the mean difference in height between married men and women within US Catholic married couples is at most 3 centimeters. Say that Alice is married to Bob, Ann is married to Ben, and so on. Then a theorist might propose to measure married US Catholic couples and compute their average heights. He asserts that the use of commonsense assumptions leads to a prediction of more than 4 cm difference. However in order to reach that conclusion he compares Alice to Ben, Ann to Bob (and so on); none of the men in his set are married to any of the women in his set. Next an experimentalist samples 1000 couples and finds an average difference of say 2.8 cm, reports the standard deviation, and claims that commonsense has been resoundingly disproved.

Do you agree with me that in such a case the mathematics of the theorist is erroneous?

Consider the set of pairs of integers {(u,v) = (n, n+1), for some n = ...-2, -1, 0, 1, 2, ...}

ie pairs of integers such that the second is one larger than the first.

Put some probability distribution over this set, ie there are probabilities p_n; nonnegative numbers adding to 1.

This defines a pair of random integers (U,V) with a joint distribution such that V = U+1 with probability 1.

Take a sample of size N from this distribution and average the U's.

Do the same again and average the V's.

Provided that the mean value of U is finite, ie sum p_n abs(n) < infty, then for large N the difference between the averages will be close to +1.

Obviously if one takes a sample of any size of pairs (U,V) and averages V-U the answer is exactly 1, whatever N.

Now if all p_n are positive the theorist can argue for for every v there is a u which is larger by 2 (namely u = v+2). ie he can pair off all the married couples such that all the women are 2 units taller than all the men, instead of all being 1 unit smaller.

Well in the Catholic church this is not allowed but perhaps the Pope will allow us to make this immoral thought experiment in our thoughts, only.

I would not say that the argument of the theorist is erroneous. I would say it is irrelevant.