To seminar 4 participants (Herman's seminar):

I am sorry about the confusion I made on the interpretation where I (at best) was imprecise. Here's a clarification.

Basically, what I meant is we are allowed to accept a higher chance of committing a type I (rejection) error. This means, as I wrote, a smaller alpha. So bottom line; we are SO sure the test of significance is rejected (meaning the coefficient IS significantly different from 0) from the evidence we have in this data. In fact, since the p-value is 0.0000, we can claim significance at very low alpha-levels.

Sorry about creating unecessary confusion, hope this clears up everything. 

Herman.