"はず" means "to be supposed to do" and it is used to emphasize an expectation that something will take place. You can use "思う" instead of "はず", but "はず" sounds more certain than "思う".
"べき" means "should do" and it is used to offer suggestions or advice from a common sense perspective. "した方がいい" is a similar expression, but it sounds less obligatory than "べき".
If you're interested in more examples of "はず", please click HERE.
If you're interested in more examples of "べき", please click HERE.
Meg and Amy are about to go to a park.
- Meg: It's nice and cool today, but did you put on sunscreen?
今日は涼しいけど、日焼け止めを塗った？ - Amy: Yes. It’s going to be hot today. The high is supposed to be 30 degrees.
[OK]うん。最高気温が30度だから、暑くなるはずだよ。
[NG]うん。最高気温が30度だから、暑くなるべきだよ。 - Meg: Okay, I will too. And I should put on a hat!
[NG] わかった、塗るね。それに、帽子を被るはずね。
[OK] わかった、塗るね。それに、帽子を被るべきね。
Ken and Leo are worried about Jim because he failed an exam.
- Ken: Jim looked a bit down because he didn't expect to fail the exam.
試験に落ちた(or不合格になった)から、ジムは落ち込んでいたみたいだな。 - Leo: Yes, but he deserved it because he played video games all the time. I think that he should study hard for the make-up exam.
うん、でもさ、いつもゲームばかりしていたから当然だよ。追試に向けて、もっと一生懸命勉強するべきだよ。
--> It's Leo's opinion or suggestion for Jim based on the fact that he didn't study hard. - Ken: I saw him going to the library a little while ago because he is supposed to study hard for the upcoming exam.
ちょっと前にジムが図書館に行くのを見たよ。次の試験に向けて、一生懸命勉強するはずだよ。
--> It's Ken's expectation based on the fact that he saw and the general way of thinking.