We can use Wilcoxon signed rank test if the paired - t test can't be used in testing paired samples. When the number of pair N is large, we can use normal approximation. The following R simulation shows that When N >= 15, the approximation is pretty good.
rankdist <- function(n = 3, k = 1e3) {
x <- 1:n
res <- numeric(k)
for (i in 1:k) {
sign <- sample(c(1, -1), size = n, replace = TRUE)
y <- x * sign
res[i] <- sum(y)
}
t <- max(density(res)$y)
hist(res, freq = FALSE, ylim = c(0, t * 1.1), main = "Barplot of the Signed-Rank Statistic", xlab = "signed-rank")
mu <- 0
sd <- sqrt(n * (n + 1) * (2 * n + 1) / 6)
xseq <- seq(-120, 120, length = 1000)
yseq <- dnorm(xseq, mu, sd)
lines(xseq, yseq, type = "l", lwd = 2, col = "red")
}
N = 5 for the top graph, N = 10 for the second, N = 15 for the bottom graph.
We can use Wilcoxon signed rank test if the paired - t test can't be used in testing paired samples. When the number of pair N is large, we can use normal approximation. The following R simulation shows that When N >= 15, the approximation is pretty good.