Apparently, whatever this "transfer rate of a high school" thing is, it seems to be a result of a binomial process and you are trying to find a model for each students "chance of a transfer", which I'll call $\theta$. Now the KISS approach would be to assume an identical $\theta$ in every school. A more sensible way would be to assume, that each school has a different $\theta$, taken out of a common distribution and the most sensible approach to assume, that each student has a different probability, which we cannot measure.
mef has written about the more sensible approach, named a hierarchical model and how you really need to learn more before you approach it. If this is just a self-study introductional example as to how Bayes works (please add self-study tag in that case), we can go the KISS way and assume, every student in the world has the same chance $\theta$.
Now it comes in handy, that there is a conjugate prior to the binomial distribution in the beta distribution.
First find a prior distribution, i. e. what do you know about $\theta$ before you look at data. Let's say, you do know, that $\theta$ is not going to be 100% and else nothing. The least informative prior to state that would be a beta distribution with $a=1$ and $b=2$.
Updating: For each student without a transfer, increase $a$ by one and for each with a transfer, increase $b$ by one:
transfers <- c(8, 12, 2, 10, 19, 3, 42, 8, 19, 11)
totals <- c(111, 108, 79, 122, 130, 95, 201, 117, 122, 103)
nontransfers <- totals - transfers
Now through the wonder of conjugate priors, our posterior distribution of $\theta$ becomes this:
curve(dbeta(x, 1+sum(transfers), 2+sum(nontransfers)))
curve(dbeta(x, 1+sum(transfers), 2+sum(nontransfers)), xlim=c(0.05,0.15))
Thus, we have build an easy Bayes-Model to start with and explain the basics. The only downside is: It is far to easy and not acceptable. With the same beta-conjugate-magic we can easily see, that not all highschools have identical $\theta$. Just plot our $\theta$ distributions for all the highschools and you can see, that our model is far from acceptable:
curve(dbeta(x, 1+transfers, 2+nontransfers), ylim=c(0,25), xlim=c(0,0.4))
for(i in 2:(length(transfers)))
curve(dbeta(x, 1+transfers[i], 2+nontransfers[i]), add=TRUE)
But maybe even this is teaching something worthwhile.