Question: How do I calculate a t-test for proportion in a small sample of 8 to do hypothesis testing?

My answer: You can do a t-test with a small sample. In the old days when there were printed tables of distributions, the t distribution was printed for df that were even smaller.

However, with a sample size of 8, any difference that is significant will pass the IOTT – the interocular trauma test – it hits you between the eyes. For example I ran this R code (anything after a # is a comment):

set.seed(1234) #Sets a random seed

x1 <- rnorm(4, 7.5, 1) #x1 is a random normal, N = 4, mean = 7.5, sd = 1 x2 <- rnorm(4,5, 1) #x2 is random normal, N = 4, mean = 5, sd = 1 t.test(x1,x2) and it was not significant at 0.05. This is a difference of 2.5 standard deviations.