Can a weak battery cause a watch to run slower?
Depending on the type of watch you have, a dead battery may or may not cause it to run slower. Some watches have a built in monitor that will make the watch run differently to signal a low battery.
Some watches, such as Tag Heuer's quartz watches, are equipped with mechanisms to alert the user that the watch battery level is low. However, mechanical watches will typically only move the seconds hand forward every two seconds rather than every second, maintaining an accurate display of time -- not actually running slower.
Digital watches do not slow down when the battery is weak; they will run at the correct time until there is not enough voltage to continue, then they stop running altogether. Only mechanical watches will slow down as the battery weakens.
A slow watch may also indicate that a circuit is shorting somewhere. It could also indicate a broken quartz crystal. A watchmaker will be able to diagnose the cause of a slow-running watch.