[Boxtaboy]

That hose is the battery acid vent tube. In the older original lead acid batteries, they connected like in the top pic below. In the newer AGM batteries, I had to remove the black connection which is too wide, and connected the see thru hose to a smaller black adapter to fit in the smaller hole in the AGM Battery (Lower pic). I find that even techs at dealers are too lazy to reconnect the vent tube when replacing batteries, so I end up doing it myself. i can see the vent tube hole in your battery in your pic above (in same location as mine in the pic below btw).