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Wikipedia states

In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.

And Britannica also states similarly

Student’s t-test, in statistics, a method of testing hypotheses about the mean of a small sample drawn from a normally distributed population when the population standard deviation is unknown.

Is the small sample size really a necessary condition for the distribution to be a t-distribution? Isn't a sample drawn from a normal population with unknown variance a necessary and sufficient criterion?

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    $\begingroup$ You’re right about necessary and sufficient. Their phrasing only describes a sufficient condition, though it would be nice if they didn’t make it sound necessary. $\endgroup$
    – Dave
    Commented Jul 4, 2020 at 14:57

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The paragraph mentions that the Student's t-test is used when the sample size is small. It doesn't mean that the Student's t-distribution requires small sample size.

When the sample size is large, the Student's t-distribution is indistinguishable to the normal distribution. And Z-test would be good enough.

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  • $\begingroup$ Strange enough, I don’t see people wanting to z-test when they have large sample sizes. I still see t-testing used. $\endgroup$
    – Dave
    Commented Jul 4, 2020 at 14:56

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