Would a paired t-test be appropriate if the variables had a range between 0 and 1? Suppose I have paired data $(X_1, \ldots, X_n)$ and $(Y_1, \ldots, Y_n)$ and I would like to compare if the means $\bar{X} = \frac{1}{n}\sum_{i=1}^nX_i$ and $\bar{Y} = \frac{1}{n}\sum_{i=1}^nY_i$ are significantly different.
The range of each variable is between 0 and 1: $X_i \in (0,1)$ and $Y_i \in (0,1)$.
In this case, would a paired t-test be appropriate due to the fact they range from 0 to 1?

As @jcken points out, if your $n$ is sufficiently large, you can utilize a t-test, but if your means are in fact proportions--and I'm guessing that they are as they're bounded between 0 and 1--and you have the counts the proportions are based on, then you're probably better off utilizing a Chi-squared test of independence (or possibly Fisher's exact test).
