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Suppose a survey contained a dichotomous variable of chief interest, such as smoker/non-smoker, along with ten other dichotomous variables. Then suppose that the incidence of smokers is a rare event with an unknown population proportion. The goal is to infer differences between smokers and non-smokers from the answers given to the other ten variables e.g. 70% of smokers are male. How would one determine the appropriate sample size (say given 95% confidence and +-5% interval)?

I'm assuming $N=Z^2 \frac{P(1-P)}{D^2}$ would not be appropriate.

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I hope this is helpful, but I'm not very knowledgeable myself.

I think a power analysis would make sense here. You enter all the variables except for one, which is calculated. In your case enter everything except for n, and leave that null. the results will tell you how many people, per group, you need to meet your requirements.

If you're using R, the code is

power.prop.test(n = NULL, p1 = NULL, p2 = NULL, sig.level = 0.05,
                power = NULL,
                alternative = c("two.sided", "one.sided"),
                strict = FALSE)

and according to the help page

n   
Number of observations (per group)

p1  
probability in one group

p2  
probability in other group

sig.level   
Significance level (Type I error probability)

power   
Power of test (1 minus Type II error probability)

alternative 
One- or two-sided test

strict  
Use strict interpretation in two-sided case

I apologize if I've made an error myself! But hopefully this is better than nothing.

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