I came across to decide a sample size for a study. I searched online and looked into a bunch of formulas for different study design and objectives, and all of them need prior information from a previous or pilot study. The study I am dealing with is new and no previous studies looked into the same objective that I am looking at. It's a proportion I am looking for, so I know for no prior information, I can use proportion as 0.5 but that's going to make my sample size larger than my capacity. I was wondering if I could use a sample size of 30, and by the Central Limit Theorem, can I run the test based on the data I will obtain? Any suggestion is highly appreciated! Thank you.
First of all, the sample size of 30 is not valid here, 30 is the sample size required for approximating the t-distribution with a normal distribution. The variable you are sampling is binomial so you can't use the $n=30$ rule of thumb.
It's not clear what you wanted to do after 30 samples. Were you going to use those 30 samples as a pilot study to then calculate a required sample size?
You are right that using $0.5$ as an estimate will give an upper bound on the required samples. Since you have on information about the true proportion that you are trying to estimate, one thing you could do is to use a Bayesian approach. Use a non-informative prior for the true proportion and calculate an expected sample size required for your test, this will give an estimate smaller than when assuming the proportion is $0.5$.
I can't see how the central limit theorem plays a role here. As said by @Hugh assume the worst case of 0.5. See the acceptable margin of error in estimating the unknown probability. In the frequentist world this is half of the width of say a 0.95 confidence interval. I would use the Wilson confidence interval. In R you can find the needed sample size by trial and error. You'll get n=96 for a +/- 0.1 margin of error and n=4*96 for +/- 0.05.
require(rms) n <- 96; binconf(n/2, n) PointEst Lower Upper 0.5 0.401924 0.598076 n <- 4*96; binconf(n/2, n) PointEst Lower Upper 0.5 0.4502388 0.5497612