I need to lower the amount of cigarettes I smoke per day.
I need to lower how many cigarettes i smoke per day.
Do both the sentences mean the same thing? Is the second one grammatically correct?
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Both of your sentences are understandable and grammatically correct, and they both mean the same thing, that is you feel the need to smoke fewer cigarettes.
However when doing less of some quantity, one would usually say
I need to decrease the amount of cigarettes I smoke per day.
I need to decrease how many cigarettes I smoke per day.
I need to decrease the number of cigarettes I smoke per day.
I need to decrease how much I smoke.
I need to decrease my smoking.
I need to cut down on my smoking.
In your example, the amount and how many is interchangeable, both mean the total quantity of cigarettes you are smoking.