I think 'B' would be better with preposition. Normally 'Home' is considered as an adverb so it doesn't take a preposition. Both are fine.
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martinleeIs there any difference between a and b?No. However, when used to modify a noun, it must be "stay-at-home" as in "a stay-at-home mom".
martinleeWhich one should I use?In this case, just using my ear as a guide, I'd say "at home". Nevertheless, both are correct.