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?
 A: Possibly appropriate, but not necessarily. There's two things to worry about: significance level, and power.

*

*If the distribution is sufficiently skew (for example), the actual significance level might not be reasonably close to the desired one at some given sample size.


*The power properties of a t-test might be relatively poor compared to one based on a more suitable choice of distributional model.
A: Yes, provided that $n$ is sufficiently large for the approximate (asymptotic) normality of $\bar{X}$ and $\bar{Y}$ to hold.
It's difficult to say exactly what value of $n$ is appropriate here, but if you make, say, a confidence interval (say, 95% or 99%) for the two population means, using the standard set up:
$$\bar{u} \pm t \sqrt{ \frac{s_u^2}{n} }$$
where $\bar{u}$ is the mean of the sample, $s_u^2$ is the sample standard deviation, $n$ is the sample size and $t$ is your relevant critical value, and neither interval spills outside of $(0,1)$, I'd imagine you would be fine
A: 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).
A: Scaling (except by zero; negative scale will be equivariant in sign and invariant in magnitude) and translating both $X_i$ and $Y_i$ by the same amount will leave the t-statistic unchanged. Something I do not like is the boundedness of these variables being used in a t-test where the probability model might be assumed to be t-distributed. Because the t distribution supports the real number line, it would seem like in principle you have the wrong probability model.
As @jcken notes, normality will approximately hold for large sample sizes and that a confidence interval based on the t-distribution might be contained in the interval $(0,1)$. But then again, such a confidence interval still assumes that the total probability mass is spread out over $\mathbb{R}$ (i.e. unbounded) rather than $(0,1)$ so such an estimated interval doesn't have the desired meaning. Perhaps an error bound could be put on it if the correct model is known, but then it isn't clear why one wouldn't just use the correct model if its corresponding interval could be calculated.
My suggestion is you either find a bounded distribution that resembles your data, e.g. possibly the Beta distribution, or that you use bootstrapping if the sample size is adequate.
