# Does the null hypothesis for a t-test have to be 0?

Is the null-hypothesis for the t-test always "there is no difference"?

It seems like, with the t-test, you always start with "there is no difference". Then you can have either of two things happen:

1. The null hypothesis is rejected. In this case, there is a meaningful difference between the two groups.
2. The null hypothesis cannot be rejected. In this case, it is inconclusive. You cannot say whether the groups are meaningfully different or not.

It seems weird to me to always start with "the two groups are the same" as your null hypothesis. But I think you are always supposed to use that null hypothesis, regardless of what you are trying to find. Is this true?

• Which t-test are you referring to? Difference between what and what? There are many variations of "the" t-test: please be specific. In the meantime, you are likely to find much of interest by searching our site.
– whuber
Oct 12, 2016 at 15:43
• The $t$ statistic looks like $\sqrt{n} (\bar{x} - \mu_0) / s$ and $\mu_0$ can be anything, it doesn't have to be zero. Oct 12, 2016 at 15:46

No, it doesn't have to be $0$. You can have your null that the difference is $5$, for example, or some other specific number (cf., here). What you can't have is a null that the difference is $\neq 0$. (For more on that distinction, see my answer here: Why do statisticians say a non-significant result means “you can't reject the null” as opposed to accepting the null hypothesis?)