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Every book I've looked on the $t$-distribution rushes through the rationale behind it. They don't even cover the thinking that gave rise to it, let alone give a rigorous derivation.

I'm looking for a book that does not short-change these questions.

(By the way, the kind of treatment I'm hoping to find would be far more extensive than what can reasonably be expected from a typical MSE answer. This is why this question is a reference request.)

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    $\begingroup$ Similar to a question I asked here, but clearly asking for references. $\endgroup$ – Antoni Parellada Aug 16 '17 at 23:58
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    $\begingroup$ There are at least two issues inherent in your Question. (1) The motivation for using the t statistic for the 't-test'. (2) The derivation of Student's t distribution as the distribution of the t statistic under $H_0.$ (3) Possibly a discussion of the noncentral t distribution to find the power of a t test. // These are usually covered in different chapters of a mathematical statistics book. Sometimes best covered in math stat books at different mathematical levels. // Maybe begin with Ch 10 (section on small sample testing for $\mu$ and $\mu_1 - \mu_2$) of Wackerly/Mendenhall/Shafer. $\endgroup$ – BruceET Aug 17 '17 at 2:49

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