Paul. A conversion rate is defined as:
$\textrm{Conversion rate} = r =\frac{\textrm{Number of goal achievements}}{\textrm{Visits}} = \frac{s}{n}$.
Assuming that the number of goal achievements is basically the number of success $s$ out of the number of trials $v$ (rather than the number of events per unit time or space), then what you are trying to do with your conversion rate data is estimate the underlying but unknown probability of conversion $\kappa$. There is absolutely no need to make a normality assumption in estimating the rate or its uncertainty. Instead, you could use the Bayesian Beta-binomial model for estimating the probability distribution of an unknown proportion.
In the Beta-binomial model, your conversion rate data $v$ follows a binomial distribution with size $n$ and probability $\kappa$:
$s \sim \textrm{Bin}(n,\kappa)$
Of course, you don't know what $\kappa$ is, so you are going to use Bayes' theorem to estimate it by combining your prior beliefs about what the probability might be and your conversion rate data. It turns out that a very useful model for the distribution of your priors beliefs about the probability in this case is the Beta distribution with concentration parameters $\alpha$ and $\beta$.
$\kappa \sim \textrm{Beta}(\alpha, \beta)$
In the Beta distribution, the parameters $\alpha$ and $\beta$ represent your prior beliefs about the concentration of successes and failures. The greater one concentration parameter is relative to another, the greater your belief that the probability favors that event. Also, the greater the sum of the concentration parameters $\alpha + \beta$, the more prior information you have about the conversion rate (say, from previous experiments using the same landing page), and the more certain you are in the expected probability $\frac{\alpha}{\alpha + \beta}$.
There is a nifty mathematical result here. It turns out that the posterior distribution of the conversion probability $\kappa$ follows a Beta distribution with the concentration parameters being a simply modification of those in your prior. The posterior distribution of the conversion probability is:
$\textrm{Pr}(\kappa|s,n,\alpha,\beta) \sim \textrm{Beta}(\alpha + s, \beta + n - s)$
That is, you just add the counts in your data to the appropriate concentration parameter ($\alpha$ being the concentration of successes and $\beta$ that of failures), and voila!
But how should you set the values of $\alpha$ and $\beta$? If you have prior information about the conversion rate for that landing page, perhaps you could set the concentration parameters to be the counts of those prior experiments. But be careful: the bigger the prior sample size, the more new conversion data you will need to overwhelm your prior beliefs. Then again, you could set the parameters so that the prior expected value $\frac{\alpha}{\alpha + \beta}$ is equal to the conversion probability from your prior experiments, but choose the parameters also so that their sum is low, reflecting your lack of information about the present landing page, say, because this landing page is much different from your previous ones. How you set the prior depends on your circumstances and your beliefs and how strong those beliefs are.
Another option is to claim ignorance about what the value of $\kappa$ might be. In this case, you could set $\alpha = \beta = 1$, which is equivalent to a continuous uniform (i.e., flat) prior distribution. Some suggest that you should use the Jeffrey's prior distribution, wherein $\alpha = \beta = 1/2$ instead, which as a U-shape with modes at 0 and 1.
Regardless of what prior distribution you choose, now you can estimate the expected value of the conversion probability, which is:
$\textrm{E}(\kappa|s,n,\alpha,\beta) = \frac{\alpha + s}{\alpha + \beta + n}$
You could also estimate the posterior variance using the formula for the Beta distribution variance, or the posteiror median, or the posteiror kurtosis, or the posterior skewness, or what have you. You could use a computer program such as R to estimate the credible range for conversion rate given your data. For example, you could estimate the posterior 95% confidence interval by firing up R and running the following code (assuming that you've defined $\alpha$, $\beta$, $s$, and $n$ in your code previously):
post_CI <- qbeta(c(0.025, 0.975), alpha, beta)
You could also use simulations from the posterior distribution to compute any number of alternative credible intervals, such as the highest posterior density interval or lowest posterior loss interval. You should seriously look into highest posterior density interval because the Beta distribution can be quite skewed, causing the traditional quantile interval approach to allow values into the interval that have lower posterior probability than values that are not in the interval. Below is the code for computing the highest posterior density interval, assuming you've defined everything as before:
sims <- rbeta(1000000, alpha, beta)
require(coda) || install.packages("coda")
post_HDI <- HPDinterval(as.mcmc(sims), prob=0.95)
