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It is commonly said that sample sizes <~30, you are unable to use a z-test for a difference of means due to the CLT. Instead it is said that your sample follows a t-distribution, and that you should use a t-test instead. The main difference between a normal distribution and t-distribution, as I have heard it described, is that a t-distribution has fatter tails, and is a bit more pointy.

But why use a t-distribution in the first place? If you want fatter tails, you can simply take a normal distribution, and adjust $\sigma$ to be lower. Why is there a need to use a t-distribution for small samples? (It sure looks an awful lot like a normal distribution)

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    $\begingroup$ The issue is that the sample standard deviation $s$ has a scaled chi-distribution rather than being the constant $\sigma$. So if the population has a normal distribution then $\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}$ also has a normal distribution but if you look at something like $\frac{\bar{X}-\mu}{s/\sqrt{n}}$ then this has a Student $t$-distribution with a different shape, and the distinction is bigger when the sample size $n$ is smaller $\endgroup$
    – Henry
    Commented Nov 4, 2021 at 16:32
  • $\begingroup$ I agree with the answer by @StephanKolassa "If you want fatter tails, you can simply take a normal distribution, and adjust $\boldsymbol{\sigma}$ to be lower." The $t$ distribution is the correct form of that "adjustment" to $\sigma$. (Incidentally, whereas $\sigma=1$ in the $z$ distribution, $\sigma=\sqrt{\frac{\nu}{\nu-2}}$ in the $t$ distribution.) $\endgroup$
    – Alexis
    Commented Nov 4, 2021 at 17:36

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You probably mean "adjust $\sigma$ to be higher". And there's the rub: by how much do you need to adjust your $\sigma$?

Yes, of course you can find a value $\sigma$ such that an $N(0,\sigma^2)$ distribution has the same critical value, or will give the same $p$ value as a $t$ distribution with $k$ degrees of freedom. But your $\sigma$ will depend on $k$ (and on whether you only want critical values, or also $p$ values - and if the latter, also on the value of your test statistic).

Conversely, we have a very well thought-through theory and actual mathematical proofs that your test statistics under the null hypothesis are indeed $t$ distributed. So let's actually use that toolset we have.

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