Yes and no. I'm sure PR is a minor contributing factor to how you rank in Google so any higher rank in the top 50 or so would increase traffic. Other than that, no.
Ive heard of plenty of high PR sites that get a very small ammount of visitors, so as roman1 said, Google factors that pr number into their ranking algo, but it is far from being directly related to the number of visitors a site gets.
One of my sites just went from PR4 to PR5 in the update that just finished. I've seen no change in the general traffic patterns at all. However, I did loose about 100 BLs as well. I expect when the affect of the higher PR gets into G via the increase in coop ads that might make a difference.
The PR that Google uses in it's algorithms is different to what we see in the toolbar as it is continually updating internally. Generally you'll get more traffic with higher PR as PR is a consequence of links and links generally bring traffic.
Well, certainly there is no relation between PR and trafic as Google does and can not track the traffic of a website. It you however mean a relation between PR and good rankings then there is a indirect one as ranking is (mostly) based on incoming links and the PR is an indicator of those links.
Sure, they can take the stats from Alexa. The real question is why on earth would they. Through the Toolbar PageRank they can easily find out which sites are more popular than others on their own. The answer to the original question is an resounding no, as compar said nicely.
Any change in visible/public PR is simply a change in reporting. Those changes are already included in search ranking, usually long before the update, so it's very unlikely that you will see any change in traffic (or SE ranking) at all.
if your pr is 0 u will get 100 visitors if your pr is 4 u will get 374 visitors if your pr is 11 u will get 15000 visitors
First of all there is no PR11. So that part of your answer is totally without foundation. Now I'd like to see the source of your information or proof for the other two assertions. I think they are both of the same quality as the PR11 assertion. Just total CRAP!