Can a paired (or two group) t-test test if the difference between two means is less than a specific value?

Question

In t-tests (paired or independent) we test if the difference in means if 0 (or another value). I'm wondering if it's possible to instead test if the difference is less than x?

Answer 1

Sure, you can do that. You don't have to test against a null hypothesis of $0$ (sometimes called a "nil null"); you can test against any value. You also don't have to do a two-tailed test; you can perform a one-tailed test (when specified a-priori). The paired $t$-test is:
$$ t = \frac{\bar x_D - \mu_{\rm null}}{\frac{s_D}{\sqrt N}} $$ Thus, to combine the two less typical possibilities noted above, you substitute your specific value for $\mu_{\rm null}$, and run a one-tailed test.

Here is a simple example (coded in R):

set.seed(2786)                     # this makes the example exactly reproducible
x1 = rnorm(20, mean=3, sd=5)       # I'm generating data from a normal distribution
x2 = x1 - rnorm(20, mean=0, sd=1)  # the true difference is 0
## this is a paired t-test of whether the difference is <1:
t.test(x1, x2, mu=1, alternative="less", paired=TRUE)
# 
#  Paired t-test
# 
# data:  x1 and x2
# t = -7.5783, df = 19, p-value = 1.855e-07
# alternative hypothesis: true difference in means is less than 1
# 95 percent confidence interval:
#         -Inf -0.02484498
# sample estimates:
# mean of the differences 
#              -0.3278085