# Paired t-test with inequality in $H_0$

I have read that paired $$t$$-test works assuming a null hypothesis with equality, i.e: $$H_0 : \mu = \mu_0$$ Is there any way to do the same test with an inequality, such as: $$H_0 : \mu \geq \mu_0$$ Or there is a (~another) statistical test to do the comparison? Thanks.

• You may want to look into equivalence testing which, essentially, reverses the classical roles of the null and alternative hypotheses. – if_the_correlations_are_zero Jan 11 at 15:48

In R with t.test using data mtcars as toy example

> t.test(x=mtcars$$mpg, y=mtcars$$mpg+rnorm(32,3,2),
alternative="less",
paired=T)

Paired t-test

data:  mtcars$$mpg and mtcars$$mpg + rnorm(32, 3, 2)
t = -9.6243, df = 31, p-value = 3.967e-11
alternative hypothesis: true difference in means is less than 0
95 percent confidence interval:
-Inf -2.468995
sample estimates:
mean of the differences
-2.996977


'Paired' does not imply a specific null hypothesis, it just means that the two measures are derived from the same sample instances.