# Difficulty understanding negative test statistic

I'm very new to statistics so I apologise if this is trivial for most of you.

I'm doing a hypothesis test where $$H_{0}: \mu_{2}-\mu_{1}=0, H_{a}: \mu_{2}-\mu_{1}<0$$

My specific problem involves seeing if there has been an in improvement using a new training regime on an Olympic Rowing Squad in 2016 and in 2020. Where 2 is denoting the 2020 squad and the 2016 is denoting the 2016 squad.

After computing my test statistic I get a result of $$-4.033$$, and then my critical value comes to $$t^{*}_{0.05}=1.708$$

Do I reject or fail to reject my null hypothesis here? This is a lower tailed test (I think) so it's confusing me since we have only done upper tailed tests in lectures. Should my basis for rejection be based on the absolute value of both or the signed value of both?

The value you get from $$t$$ table is for the upper tail. For the lower tail, it is $$-1.708$$ and since your value is to the left of it, you'll reject the null hypothesis.
• Not exactly. You just make sure which $t$ value you need. For $<0$ hypotheses, the significant portion is the left tail, so $t^*=-1.708$. If you take absolute values and compare, for a test statistics $4.033$ (instead of the negative), you'd reject the null hypothesis. However, your goal was to reject when $\mu_2-\mu_1<0$. – gunes Feb 23 at 12:31
• I still don't quite understand. So because $-4.033<-1.708$ I am rejecting the null hypothesis? Shouldn't I be failing to reject if the test statistic is less than the critical value? – Karl Coogan Feb 23 at 12:38
• Don't memorise it. If hypothesis seeks significance in upper tail, i.e. $H_A: \mu_2-\mu_1>0$, you reject null when it's larger. If in lower tail, you reject null if it is smaller, if two-tailed, you reject null when absolute is larger. Visualize a bell curve centered around $0$, and mark $-1.708$ as your border and mark your test statistic as well. The question is "is it outside (tailwise) the border "? – gunes Feb 23 at 12:42