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.
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.