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Tuesday, January 03, 2006

Are my football predictions accurate?

First, I'll mention one problem that I am working on over this break. Consider the standard plane in two-space. Then in, say, the first quadrant, consider the "wedge" shaped region W bounded by a line through (0,0) of slope m, where m > 0, and the x-axis. Let X denote some infinite set of points in W which has (0,0) as a limit point. Clearly it is possible to find a single variable function f whose graph runs though an infinite subset of W such that
  • f is continuous on the entire real line
  • f is differentiable except for possibly at x = 0.

Here is how one can obtain such an f : Select from W some infinite subset {xi} with the following property: r(i+1)where {r(j)} is some strictly decreasing sequence which converges to 0. Next, construct an polygonal infinite arc by connecting the points xi to each other via line segments, and then setting f(0)=0, and then choosing a real number sequence ti where ti converges to 0 and then setting xi = (ti, f(ti)). Clearly such an f is continuous. To make f differentiable: let yi denote the midpoint of the segment connecting xi to x(i+1). Then do a three point spline smoothing process to obtain a smooth segment through y(i-1), xi, y(i) such that the smooth arc has derivative equal to the slope of the segment containing yi for each i. This modification makes f smooth, except for possibly at t = 0
So the question is: how does one make f differentiable from the right at t = 0 ; that is, I want {limit as dt goes to zero) of (f(0+dt)-0)/dt to exist. The type of thing I am worried about would be things like f(t) = tsin(1/t) which has a limit as t goes to zero, but cannot be made to be differentiable there. I have some rather cumbersome ideas to try out over the next 4-5 days.

Now, about football predictions: I made some bowl game predictions here:

and my record was 17-9 straight up (on correctly picking the winner) and 15-10-1 (on correctly picking the winner vs. the published point spread). Example: at the Insight Bowl, Arizona State was favored to be Rutgers by 11.5 points. I picked Arizona State to win (which they did) but I picked Rutgers to cover ( that is, either win or lose by no more than 11 points). The final score was Arizona State 45, Rutgers 40, so I won "straight up" and also won "against the spread". On the other hand, Michigan was an 11 point favorite against Nebraska in the Alamo Bowl. I picked Michigan to win, but Nebraska to cover. Nebraska won 32-28, so I missed the "straight up" pick, but won the "spread" pick.

So, the question obtains: am I a good at making football predictions? I define "good" as being "better than chance". In statistics, one can safely say that they are better than random chance if the probability of, say, a coin flip doing as well is less than or equal to 5%.

So I decided to do a hypothesis test:

Ho: my predictions are no better than a random coin flip

Ha: my predictions are better than a random coin flip.

I used the binomial distribution (26 bowl games, a coin flip has .5 probability of being correct each time, so how many times will a coin be right?). In the days of old, it was usual to use the normal approximation to the binomial or to use tables; but now-a-days there are handy calculators such as this one:

So if X is the random variable that denotes the number of times a coinflip would be correct out of 26 trials, we see that P(X>= 17) = .0843 and P(X>=18) = .0378. Now we see that x = 15 for the spread betting and x = 17 for the straight up betting (note: my point spread picks were correct 15 times and P(X>=15) = .2786. ) So that tells us that in neither case (straight up or by the spread) gives convincing statistical evidence that I can do better than a coin flip, though I was "close" to being better than a coin flip with my "straight up" predictions.


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