I don't quite understand the assumptions of the t-test Wikipedia gives. I'm currently taking an intro statistics course, and when we covered t-tests, we learned the assumptions for a t-test are that:

- The population is assumed to be normally distributed
- Samples are random
- If there is deviation from either of the above, sample size must be larger.

Are these equivalent to the three that are listed on Wikipedia?

Assumptions given by Wikipedia:

The assumptions underlying a t-test are that

- $X$ follows a normal distribution with mean $mu$ and variance $sigma^2$
- $s^2$ follows a $χ^2$ distribution with $p$ degrees of freedom under the null hypothesis, where $p$ is a positive constant
- $Z$ and $s$ are independent.

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#### Best Answer

*What makes something an assumption?*

In order to work out the sampling distribution of the test statistic under the null hypothesis, some assumptions are required. In the case of the "standard" t-statistics (the ones that actually have t-distributions if the conditions hold), if you have independent, identically distributed normal observations (or iid normal pair-differences in the case of a paired test), then the t-statistic will have a t-distribution.

Neither of the lists you give quite work as a list of assumptions for the t-test.

- The population is assumed to be normally distributed
- Samples are random

okay so far

- If there is deviation from either of the above, sample size must be larger.

This is not an assumption of the t-test — it's advice about what you need if the assumption of normality isn't met (it's of no use if assumption 2 doesn't hold though), and then it's not $t$ in any case; you'd have to invoke two results (CLT + Slutsky) to argue the statistic would be asymptotically normal.

- $X$ follows a normal distribution with mean $mu$ and variance $sigma^2$

Well you actually also need independence. Once you add that (which is related to your "random sampling" assumption), you're set.

- $s^2$ follows a $χ^2$ distribution with $p$ degrees of freedom under the null hypothesis, where $p$ is a positive constant

This is a consequence of the first assumption (once you add the missing independence)

- $Z$ and $s$ are independent.

This is also a consequence of the first assumption (again, once you add the missing independence)