If you look at Mike and Phil's examples, you will see that in those cases," a majority x" means "the x of the majority". If you can use it that way, then it works. Rules, votes and opinions are things that a majority of people can have. Your examples "majority family" and "majority people" don't work like that as far as I can see. Just because you found them does not necessarily mean that they are proper English! Feel free to give me a link so I can see if they can in fact be used that way. Having said that, they could work if 'majority' was qualified, such as 'ethnic majority family' - meaning a family of people of (belonging to) the ethnic majority.

In English, we can usually use a noun to modify another noun. For example: "the majority opinion." An alternate construction is with a prepositional phrase "the opinion of the majority." "The Department of State" or "The State Department."

The only example I can think of off of the top of my head uses 'majority rule' or 'majority vote': For example: "In that country decisions are made by majority rule." This means that decisions are determine by the will of majority of the people. In this context majority = more than one half