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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?

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    $\begingroup$ 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. $\endgroup$ – whuber Oct 12 '16 at 15:43
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    $\begingroup$ 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. $\endgroup$ – dsaxton Oct 12 '16 at 15:46
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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?)

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It should be some value you specify before you have any data ... but it doesn't have to be zero.

For example, imagine someone makes a claim that their tutoring program improves mathematics test scores by 5 (out of 100). You could make that 5 improvement your null value in a paired test and then do an experiment where you measure scores, give the training and then measure again after (though ideally you'd also have a control group that you just did the before and after tests on to account for the possibility that the testing alone improved the scores).

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