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I have to answer the question: does the success rate of quitting smoking depend on the treatment method? for the data pictured.

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what is the appropriate hypothesis test for this kind of data? i understand comparing two proportions but this has got me confused. thank you in advance!

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You might use a $\chi^2$ test for homogeneity of the three subpopulations (Nicotine gum, Nicotine patch and Nicotgine inhaler). I understand that you have taken three different samples. If you have taken a single sample (that is, without fixing the number of persons taking each treatment: they just happened to fall in those categories) then the test would be one of independence.

Either way, since inference is conducted conditional on the margins, the test statistic (and conclusion) is the same. If you are an R user, function chisq.test will give you the result easily.

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Chi-square test of independence: http://www.biostathandbook.com/chiind.html

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