Nate Silver’s Models and “expected value” versus “predicted value”, confidence intervals, p-values, etc.

Workout notes I ran in the morning (already recorded) and lifted over lunch:
rotator cuff
pullups: 15, 4 sets of 10
super sets of rows, curls and pull downs, 3 sets each: rows: 10 x 210 (Hammer machine, last one w/ rotated grip), pull downs: 10 x 165, last one with rotated grip, curls: 10 x 52.5 pulley
bench press: 10 x 135, 7 x 165, 6 x 170, 5 x 175 (did ab crunches to rest)
super sets: 2 sets of military presses, incline: incline: 8 x 135, 7 x 135, military: 15 x 45 dumbbells (supported)

I tire out too easily; I really thought I could do more sets of 15 of pull ups.

Mathematics and Physics
It looks as if the mathematics of bifurcating trajectories will enable physicists to make progress on unifying quantum mechanics and relativity theory.

I’ll have to take a look.

Statistics and the 2012 Presidential Election
Yes, I’ve been following the “horse race” very closely and think that I have some stuff that I can use to explain things to students.
For one: have studied “confidence intervals” and “hypothesis testing”. Nate Silver’s recent article has some examples of these:

First of all: what are these?

Each one of these is a so-called “90 percent confidence interval” that shows “Obama’s true support” in these states (or Congressional districts, in the case of NE-2 and ME-2).
What this means: we are 90 percent certain that Obama’s true support falls somewhere in this interval. Example: in New Mexico, Obama’s lead is 2.3 to 15.5. In Montana, Romney’s lead is between 2.2 and 17.1.

Note: the widths of these intervals are a bit different; that is because the respective distributions have different “standard deviations” and the higher the number of people polled, the smaller the standard deviation. A 90 percent confidence interval is about plus/minus 1.645 standard deviations.

Now note that each interval is colored blue, red or mixed. An “all blue” band means that we are 90 percent sure that President Obama leads in that region. An “all red” means that we are 90 percent sure that Governor Romney leads. If a band is mixed color (Wisconsin to North Carolina (mostly)) that means that we do NOT have 90 percent confidence that Mr. Obama or Mr. Romney leads.

This is an example of “hypothesis testing”; if a band is all blue we reject the “null hypothesis” that the race is tied and conclude that one candidate is ahead with 90 percent confidence.

However, unless the dividing line of the colors is right in the middle of the band, we can make a probability estimate of who is ahead.
Let’s look at Wisconsin. We see just a tip of red there and a number that says 88 percent. What this means: if we were willing to settle for being 88 percent confident, we could concluded that Obama was ahead there. In North Carolina, if we wanted to settle for 81 percent confidence, we’d conclude that Romney was ahead.

Now write these percentages as decimals and subtract them from 1. That is called “the p-value”. For Wisconsin: we’d have 1-.88 = .12 and we’d say “P = .12” for a “one-tailed test”.
For North Carolina: 1-.81 = .19 so we’d say P = .19.

Predicted values versus expected values
Nate Silver also says this:

Mr. Obama is not a sure thing, by any means. It is a close race. His chances of holding onto his Electoral College lead and converting it into another term are equivalent to the chances of an N.F.L. team winning when it leads by a field goal with three minutes left to play in the fourth quarter. There are plenty of things that could go wrong, and sometimes they will.

But it turns out that an N.F.L. team that leads by a field goal with three minutes left to go winds up winning the game 79 percent of the time. Those were Mr. Obama’s chances in the FiveThirtyEight forecast as of Wednesday: 79 percent.

First about that NFL stat: if that sounds strange, let’s remember that the 79 percent is the probability that the team that is down by 3 with first and 10 at its own 20 with 3 minutes to go in the game loses the game. That doesn’t mean “never catches up”; they could catch up, and even go ahead and still lose the game. This is “total probability of losing the game.

Another way of seeing it: this is like an average NBA player taking a free throw; if they make it, Obama wins. If they miss, Romney wins.

That puts it into some perspective. The reason: we are trying to predict the outcome of THIS single election. That is a “predictive value” problem.

Now if we were having this election in, say, 1000 parallel universes with roughly the same conditions, Obama would win close to 80 percent of such elections. This would be an “expected value”: the percentage of Obama wins over a large number of cases with similar conditions.

So if the election were decided by a “majority of election outcomes over a large number of trials”, well, this election WOULD be over and THAT would be an “expected value” problem.
But this election is valid on this universe only, and that is a predictive value problem. Hence BOTH campaigns are sweating at the moment.

November 1, 2012 - Posted by | 2012 election, mathematics, physics, politics, poll, statistics, weight training

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