First of all compare these two English sentences.
|He should have arrived by now.|
|You should quit smoking.|
In the first sentence, you are drawing a tentative conclusion; in the second, you are giving advice or a recommendation to someone. In Japanese this distinction is important as it determines the appropriate word we need to use to express “should” in English.
はず indicates a conclusion based on your current knowledge of the situation or a result that you believe is natural consequence of events.
|He should have arrived already.|
|You ought to know that running a red light is illegal.|
You are are presuming that such knowledge is common sense.
|She just came back a moment ago so she should be in her room.|
To modify nouns the possessive is required.
|You should know that without having to ask.|