Yeah, I doubt a jumping spider would bite. For one they're usually like a couple millimeters long, and so most probably can't even penetrate the skin. But more importantly, you aren't an organism or predator to them, you're a piece of terrain. You're bigger to them than the stone giants from The Hobbit movie. It isn't in their best interest to simultaneously piss off and inform such a creature where on its body they are.
Even among the much maligned spiders such as black widows and brown recluse, they generally do their best to avoid people. The main bite cases are either when rummaging around in dark, undisturbed areas where they're hiding, or small children deciding that, since it moves, it is a toy to be played with. The latter case being the most likely to cause death. Primarily because the venom typically is only fatal for young children or those who are health-compromised in other ways.
Considering spiders live in pretty much every home, they're much safer than dogs, which kill about 5 times more people a year in the US, despite living in only a fraction of homes.
Even with the Super Scary Deadly Brown Recluse, you pretty much won't get bit. Despite that, according to articles like this, hundreds of brown recluse bites are diagnosed by doctors each year.... in places they don't live and have never been seen.
http://newsroom.ucr.edu/304Overdiagnosis of brown recluse bites is a nationwide problem. In 1990 in South Carolina, 940 physicians reported 478 brown recluse bites. In 2000 in Florida, 95 brown recluse bites were reported from the 21 counties under the jurisdiction of the Tampa Poison Control Center. Yet arachnologists who have worked for years in these regions and have collected thousands of spiders, have never found recluses, and homeowners have yet to submit a local brown recluse to them for verification.
http://spiders.ucr.edu/myth.html