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Let's say I know the population standard deviation, but the sample size is small (≤30). Can I use the z-test? The reason I ask is that I see two different statements.

  1. We can use the z-test, if we know the population standard deviation AND the sample size is >30.
  2. As long as we know the population standard deviation, we can use the z-test. The sample size does not matter (above or under 30).

What do you think and why? Thanks.

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    $\begingroup$ Yes if $X \sim N(\mu,\sigma^2)$ for known $\sigma$. (Is it plausible to know the population standard deviation but not the mean?) But if it has another distribution then it depends on whether a normal approximation would be close enough: in some cases it may be, while in others not $\endgroup$
    – Henry
    Commented Nov 28, 2022 at 0:18

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What is the hypothesis here? What does $X$ follow?

If $X\sim\mathcal N(\mu, \sigma^2) $ with $\sigma$ being known, and $\mathcal H_0: \mu=\mu_0, $ then, of course, one can resort to the test statistic $$z:=\frac{\bar x-\mu_0}{\sigma/\sqrt n}.\tag 1$$

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  • $\begingroup$ Yes, that's exactly the hypothesis I was referring to. So it does not matter even if n≤30, correct? Thanks. $\endgroup$
    – Fred Chang
    Commented Nov 28, 2022 at 0:43
  • $\begingroup$ This is an exact test. So, yes. $\endgroup$
    – User1865345
    Commented Nov 28, 2022 at 0:44
  • $\begingroup$ The question does not assume a Normal population distribution. $\endgroup$
    – whuber ♦
    Commented Nov 28, 2022 at 1:31
  • $\begingroup$ @whuber that's why I asked at the outset. What the specifics are. It seems it is indeed the case as OP said above. $\endgroup$
    – User1865345
    Commented Nov 28, 2022 at 1:46
  • $\begingroup$ I suspect the OP might not realize you made this extra (very strong) assumption. $\endgroup$
    – whuber ♦
    Commented Nov 28, 2022 at 2:03

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