Haselton needs an article about basic statistics. The 95% confidence interval on the difference between the two proportions is 6% +/- 17%, i.e. the range from about -11% to +23%. This (a) demonstrates that the sample is indeed underpowered to distinguish the sort of effect sizes that Haselton appears to be interested in, and (b) demonstrates that a +20% difference in proportion, contrary to Haselton's assertion, absolutely falls within the range of true values that can't be ruled out at a standard level of statistical confidence given the outcome of this experiment.
see http://www.kean.edu/~fosborne/bstat/06d2pop.html for the basic statistics.