The difference between two population means

Question

I am trying to answer this question:

There is no difference between two population means. From each of these two populations
a sample is selected and then for both samples the means are calculated. 

Consider this:
"since the null hypo is true, a significant result can never be found when we perform
a t-test for independent measures (two-tailed, alpha = 0.05)

Can we say that is statement is in/correct, partially correct, or insufficient to answer? I am really unsure of which path to take.

Answer 1

This is totally false. Let's do an R simulation.

set.seed(2021)
ps <- rep(NA, 1000)
for (i in 1:1000){

    # Simulate data from two identical distributions
    #
    x <- rnorm(100)
    y <- rnorm(100)

    # Test the means, save the p-value
    #
    ps[i] <- t.test(x, y, var.equal=T)$p.value
}

hist(ps)
summary(ps)

When you run this, you will notice that a number of the p-values fall below $0.05$. In fact, about $5\%$ will fall below $0.05$.

It turns out that, under the null hypothesis, the p-value has a uniform distribution on $[0,1]$.