t-test when the data population is not normally distributed

Question

I understand that according to CLT the estimator would become normal but, If my population is not normally distributed, let's say it is uniformly distributed. Can I use t-test? let's also assume that I have a large sample size more than 1000.

Answer 1

Yes, you can, for precisely the reason you give: even if the underlying population is not normally distributed, the mean (or more precisely the difference between the means) is asymptotically normal. (There are some conditions on the underlying populations that are usually satisfied in the real world, and certainly for underlying uniform distributions.)

Let's illustrate with a simulation (R code): we consider two populations, one $U[0,10]$ and the other $U[0.5,10.5]$, and a total sample size of 1000, half from each population. Here is a sample and a t-test:

nn <- 1000

draw_1 <- function(n) runif(n, 0, 10)
draw_2 <- function(n) runif(n, 0.5, 10.5)

set.seed(1)
sample_1 <- draw_1(nn/2)
sample_2 <- draw_2(nn/2)

t.test(sample_1,sample_2)

which yields

        Welch Two Sample t-test

data:  sample_1 and sample_2
t = -3.1827, df = 996.74, p-value = 0.001504
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.9387957 -0.2226748
sample estimates:
mean of x mean of y 
 4.956549  5.537284

Now, to see that the difference in means is normal enough, we simulate drawing samples and calculating means many times:

means <- replicate(1e4,{
    sample_1 <- draw_1(nn/2)
    sample_2 <- draw_2(nn/2)
    mean(sample_2)-mean(sample_1)})

hist(means)

Of course, this difference is not really normal (for one, it's bounded between -9.5 and 10.5, whereas the normal distribution is unbounded), but it's normal "enough" for the t test to work.