iPhone 5s lost 10% battery?

I just received my iPhone 5s yesterday in the mail through US Cellular. The first concern I had is that the phone was dead right out of the box. I thought that normally they had at least half battery when first powered up. Anyway, I plugged it into my computer and it took 20 minutes of charging before it was able to turn on! I don't know about you, but that seems way too long for it just to turn on. After setting the phone up and upgrading to iOS 9, it was charged in 1.5 - 2 hours. I used it right before I went to bed (just surfing the web and watching a couple YouTube videos) and the battery went down to 97%. Since I don't have it activated yet, I shut it down until tomorrow when I got home from school. Well, I just powered it back on (it's been about 15 hours since it was shut down), and the battery was at 86%! It should not drop that low if the whole phone was turned off. Maybe 2% at the most. Do you guys think I have a bad battery and should exchange for a new phone? I had Apple run a diagnostic test on it through online chat and they said it was good. Not too sure about this... Any thoughts? Thanks.