"Half an hour" versus "half hour"

You could say "a half hour" or "half an hour". The article choice depends on where in a phrase you place it.

Would you ever say "I'll be back in hour"? No, because while possibly legal, the use of the article "an" would be needed to make it sound like a native English speaker.

"half-hour" is commonly written with a dash (hyphen) and is a distinct word in the dictionary. So, "a half-hour" = 30 minutes = "half an hour".

In Australia we say 'half an hour'. You might hear 'a half hour', but it would sound stilted. This is probably changing with the influence of American television.

The top one is correct.

* I'll be back in half hour

would be OK if it said

I'll be back in a half hour

Note that "a half hour" is an American usage, British people only use "half an hour".