Clarification on t-test assumptions

Question

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:

1. The population is assumed to be normally distributed
2. Samples are random
3. 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.

Answer 1

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.

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

1. The population is assumed to be normally distributed
2. Samples are random

okay so far

3. 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)

Answer 2

The two sets of assumptions are not identical, but do overlap. Here are the assumptions that are relevant to t-tests on means (not all t-tests are on means - e.g. a test of correlation)

1) Observations are random variables that are independent and identically distributed (iid).

This is partially addressed in your class's assumption number 2 (the random part). The random part is not clearly stated in the wikipedia list, though see below regarding identically distributed.

2) The observations are from a normally distributed population.

That the population is normally distributed is listed in both sets. The subtle difference between the two is that the wikipedia version states that there are fixed parameters of mean and variance - implying that the population mean and variance are constant from one observation to the next (identically distributed - as per point 1). This is probably meant to be implied but is not explicitly stated in your class's equivalent assumption.

3) If more than one population is sampled from, then the populations have equal variances (homogeneity of variance).

This is not stated in either list. In the case of the wikipedia example it could be because the formula is written for the single-sample case. You could consider this simply an extension of point 1, but I often see it separately listed.

OTHER STUFF

A) The third assumption you list for your class stuff is partly incorrect. While it is true that larger samples provide some robustness against non-normality, it's not true that they protect against biased sampling. If your observations are not a random sample from the population, I don't think you can fix the problem by taking a larger sample. In any case, the fact that larger samples may be better in some circumstances is not an ASSUMPTION of the test. At best it's a property.

B) Wikipedia's second and third assumptions are ones I haven't seen before. I believe that these are insured in the case of the t-tests on a mean by virtue of the other assumptions I listed (in particular the chi-squared nature of s-squared arises from the assumption of the normality of X in the means tests).