Paired T Test Hypothesis Interpretation

Question

With a paired T test I understand the first part of the hypothesis:

$H0: \mu_d = 0$

The alternative hypothesis I get a little mixed up and I am trying to figure out when to appropriately use the alternatives.

If the alternative hypothesis is:

$H1: \mu_d\neq0$

I think this means we are just using the paired T test to see if there was a change between test 1 and test 2.

If the alternative hypothesis is:

$H1: \mu_d>0$

Is this setting up that the alternative hypothesis is checking that the results of test 2 were lower than that result of test one?

If so then by extension an alternative hypothesis of:

$H1: \mu _{d} < 0$

This is checking that test 2 had higher results then test 1?

Any insight would be greatly appreciated.

Answer 1

That depends entirely on how the test statistic is specified, or equivalently, how $\mu_d$ relates to the means of each group. If you define $\mu_d = \mu_2 - \mu_1$, that is, the mean of the second tests minus the mean of the first tests, then $\mu_d > 0$ implies $\mu_2 > \mu_1$; conversely, if $\mu_d = \mu_1 - \mu_2$, then $\mu_d > 0$ implies $\mu_2 < \mu_1$. Correspondingly, if you use the first definition, your test statistic must be constructed as $$T = \frac{\bar x_2 - \bar x_1}{s/\sqrt{n}},$$ whereas if using the second definition of $\mu_d$, the order of the difference must also be interchanged.