Math Puzzle!

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OK kids, here's the deal. An economist who shall remain nameless (but he writes for the NYT) thought he blew up the theory that the recession was due to a misallocation of labor across sectors, because the change in unemployment from Dec-07 to Dec-08 was high in several states that didn't experience a big housing bubble.

I thought that was an odd measure, and so I ranked states (A) by their change in unemployment rates from Jun 06 to Dec 08, and then I ranked the states again by (B) their percentage change in house prices from 2q 06 to 4q 08.

I found that 5 of the top 6 states in column (A) were in the top of column (B). So that seems like a very strong relation.

Can someone talk me through how I would compute the chance that this just happened randomly?

For extra credit, how do we deal with the fact that after I saw the results, I came up with the best way to describe them? In other words, I originally looked at the top 10 states in both lists. But since there were no more matches in the 7th through 10th slots, I just dropped them and reported "5 out of the top 6 match." So I'm thinking this cherry picking after the fact may be contributing to the impressiveness of my result, but maybe not?

Comments

  1. Bob Murphy12:12 AM

    I am leaving a comment here so that my email will be tied to this post. I wait with shrinking hope that Wabulon will show me the way.

    ReplyDelete
  2. Anonymous1:23 PM

    Odds that #1 is in one of top 6 spots: 6/50
    Odds that #2 is in one of remaining 5 top spots: 5/49
    Odds that #3 is in one of remaining 4 top spots: 4/48
    Odds that #4 is in one of remaining 3 top spots: 3/47
    Odds that #5 is in one of remaining 2 top spots: 2/46

    100% x (6/50)(5/49)(4/48)(3/47)(2/46)=0.0002832% (1 in 353,127).

    Generalization: odds that the top A of C values are in the top B of C spots is:

    (B! / (B-A)!) / (C! / (C-A)!)

    So the odds that the top 5 were in the top 10 spots is:

    (10! / 5! ) / (50! / 45!) = 0.0119% (1 in 8408).

    Math checks:
    Odds that the top 1 is in the top spot is (1!/0!) / (50!/49!) = 1/50, as expected.
    Odds that the top 50 are all in the top 50 spots is (50!/0!)/(50!/0!) = 1, as expected.
    (note: 0! is defined to be equal to 1.)

    I'm not promising this is right, but it looks good to me.

    Wabulon?

    ReplyDelete
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