A group of men and a group of women take a test which reveals that the average score of the women ...

A group of men and a group of women take a test which reveals that the average score of the women is higher than that of the men.  The experimenter wants to be very sure that this difference is statistically significant before claiming that women have better abilities than men and so sets the alpha (α) level for the statistical test to .001 rather than the more typical .05.  Do you see problems that may arise with this?