## Paul the Octopus: Credible? July 11, 2010

Posted by tomflesher in Economics.
Tags: , , ,

Paul the Octopus (hatched 2008) is an octopus who correctly predicted 12 of 14 World Cup matches, including

Spain’s victory over the Dutch. Is his string of victories statistically significant?

First, I’m going to posit the null hypothesis that Paul is choosing randomly. As such, Paul’s proportion of correct choices should be .5 ($H_o : \bar{p} = .5$). His observed proportion of correct choices is 12/14 or .857.

The standard error for proportions is

$\sqrt{\frac{p(1-p)}{n-1}} = \sqrt{\frac{.857(.143)}{13}} = \sqrt{\frac{.123}{13}} = \sqrt{.009} = .097$

The t-value of an observation is

$\frac{p}{se} \sim\ t_{df} = \frac{.857}{.097} \sim\ t_{13} = 8.84 \sim\ t_{13}$

According to Texas A&M’s t Distribution Calculator, the probability (or p-value) of this result by chance alone is less than .01.

Using the binomial distribution with $\lambda = .5$, the probability of 12 or more successes in 14 trials is a vanishingly small .0065.

So, is Paul an oracle? Almost certainly not. However, not being a zoologist, I can’t explain what biases might be in play. I’d imagine it’s something like an attraction to contrast as well as a spurious correlation between octopus-attractive flags and success at soccer.