SAN RAMON, Calif. (AP) — Google’s next internet-connected home device will test whether consumers are willing to trust the company on a new frontier: monitoring their sleep. That technology will be a key feature on Google’s newest Nest Hub, a 7-inch smart screen unveiled Tuesday. The $100 Nest Hub offers the same picture, video and voice-activated assistant as the previous version, plus sleep monitoring that doesn’t require people to wear a device in bed. Google promises the feature was carefully designed to respect people’s privacy, but some may have doubts given the company’s long history of online surveillance to help sell ads.