How to model a skewed Student's t disribution I have a small number of samples (5) of a large population (~10,000). The samples are percentages and hence I know from the context that no answers are possible below 0% or above 100%. From this one set of samples I've calculated my t statistic and confidence intervals; however, for anything above 95% confidence my interval width is such that the predicted upper limit is above 100.
My understanding is that is you have specific knowledge like this you can create a skewed Student's t distribution. First, is this correct? Second, how would this apply to the upper and lower limits of my confidence interval? I understand that two values are important in calculating this, the skewness and kurtosis, but I'm not sure how to use them. 
My skewness value: -1.82
My kurtosis value: 3.448
Sample Values:
79.465
91.905
91.096
88.144
90.599

 A: Here's a simple Monte Carlo simulation. Assuming that the population distribution is Beta with parameters 80 and 1.1, I draw samples of 5 observations, then plot the histogram of their mean.
x=mean(100*betarnd(80,1.1,5,100000));
subplot(2,1,1)
plot([94:0.01:100],betapdf([94:0.01:100]/100,80,1.1))
title 'Population distribution'
subplot(2,1,2)
histfit(x,40,'t')
title 'Sample mean'


You can see that the sample mean does not fit neither normal nor t-distribution, so t-statistics would be useless. I don't think that you can derive the distribution of the sample mean without knowing something about the distribution of the population beyond its bounds. Your sample variance is 26, i.e. too high relative to the size of the population and its bounds in order to apply CLT here or other asymptotic analysis reliably.
Now if instead of assuming some random population distribution, let's fit Beta distribution to your data.
xp=[79.465
91.905
91.096
88.144
90.599]

% fit Beta distribution to the sample
ab=betafit(xp/100)
% generate many samples of 5 observations
x=mean(100*betarnd(ab(1),ab(2),5,100000));

subplot(2,1,1)
plot([0:0.01:100],betapdf([0:0.01:100]/100,ab(1),ab(2)))
title 'Population PDF'
subplot(2,1,2)
histfit(x,40,'tLocationScale')
title 'Sample mean distribution'

phat=mle(x,'distribution','tlocationscale')
pd = makedist('tLocationScale','mu',phat(1),'sigma',phat(2),'nu',phat(3))
disp 'Conf. intervals of population mean'
pd.icdf([0.025 0.975])

This gives an output:
pd = 

  tLocationScaleDistribution

  t Location-Scale distribution
       mu = 88.2223
    sigma = 1.82489
       nu = 89.9698

Conf. intervals of population mean

ans =

   84.5968   91.8478


The 95% confidence interval seems reasonable, but the t Distribution fit is not so good, it's not skewed as the sample mean distribution. You could do something along this line, so long that you have a good idea about the population distribution. I still don't tink that the confidence interval would be reliable given the tiny size of the sample and its high variance.
