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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.

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  • $\begingroup$ You may want to look into equivalence testing which, essentially, reverses the classical roles of the null and alternative hypotheses. $\endgroup$ – if_the_correlations_are_zero Jan 11 at 15:48
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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
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'Paired' does not imply a specific null hypothesis, it just means that the two measures are derived from the same sample instances.

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