Final answer:
It is true that both common carriers and innkeepers are held to a very high degree of care and are liable for slight negligence. This high standard is because they offer services to the public and are responsible for their customers' safety. The information about proprietary colonies is incorrect; proprietors had multiple responsibilities beyond profit collection.
Step-by-step explanation:
The statement that common carriers and innkeepers are liable for slight negligence is indeed true. In legal terms, common carriers (like airlines, buses, and railroads) and innkeepers (like hotels) are required to exercise a high degree of care towards their customers. This means that they must take all reasonable precautions to ensure the safety and well-being of their patrons. Failure to do so, even if the negligence is slight, can result in liability for any harm that results.
In contrast to the initially presented information regarding proprietary colonies, it is false that the proprietors of a proprietary colony had no responsibilities except to collect profits. In reality, proprietors had a range of obligations including governance, land distribution, and upholding the laws and interests of the Crown or state entity from which they derived their authority.