confident proportion I conduct an experiment and I get #installs=2 out of #clicks=100, I am interested in the proportion #installs/#clicks. Here, I get ratio to be 2%.
I conduct another experiment and get #installs=200 out of #clicks=10000. Again, the proportion is 2%.
I was wondering which scenario gives better representation of the ratio in the population. What are the mechanical/statistical ways to determine this?
(Apologies if I am not able to articulate the question accurately; I would appreciate if someone can help me in arriving at right questions in this scenario)
 A: You can do this via the Beta distribution. The probability density in the case of 2 out of 100 is $Beta(1+2, 1+98)$, whilst the one of 200 out of 10000 is $Beta(1+200, 1+9998)$. A plot of both shows, how much more precise the latter examination is:

Or if we look at just the first tenth of that x axis:

On the x-axis we have all possible "true" ratios and on the y the densitiy of the probability of this ratio. Areas under the curve depict the probability of the true ration being within a certain intervall. Because you have more data in the second case, you can make a more precise definition of the "true value". This is a so called Baysian approach with a "flat prior" (make shure to understand that concept before using this thing".
A more Frequentist way to look at the picture would be a confidence interval of the true ratio. Whatever a condfidence interval acutally is, it can be computed like this in R. However, be shure to understand what a confidence intervall truely is and not what people tend to think, what it is, before using this thing.
> binom.test(2, 100)$conf.int
[1] 0.002431337 0.070383932
attr(,"conf.level")
[1] 0.95
> binom.test(200, 10000)$conf.int
[1] 0.01734654 0.02293792
attr(,"conf.level")
[1] 0.95

Edit:
I made a mistake above. In the 200:9800 case, the Beta-Distribution is obviously $Beta(1+200, 1+9800)$ and not, as I wrote above, $Beta(1+200, 1+9998).$  I did not correct the plots above as it will not make much difference to the plots and nobody is going to read actual numbers from the plots above.
