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I've been trying to formulate this question and have struggled, so if it seems ill-worded, I apologize:

A statistics book I have has a table in the back that says:

  • For an inference test regarding a proportion, generally a z-test is used.
  • For an inference test regarding a mean, a t-test is generally used.
  • For an inference test regarding “goodness of fit”, a chi-square test is generally used.
  • For an inference test regarding association between two quantitative variables, a t-test is used.

How are these decisions made? How can I know that a “goodness of fit” test statistic is optimally calculated by a chi-square statistic, etc.

Additionally, how is it that the normal distribution somehow fits the proportion (z) statistic, but not a t-distribution, or chi-square distribution, etc.?

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  • $\begingroup$ Welcome to Cross Validated! Good question, but I fear it's rather broad for this site, as the usual answer spans a chapter or several in books introducing statistical theory. $\endgroup$
    – Scortchi ♦
    Commented Apr 2, 2016 at 18:29
  • $\begingroup$ @Scortchi I see. Any good references to start? $\endgroup$
    – ZachTheRiah
    Commented Apr 2, 2016 at 18:35
  • $\begingroup$ See here: stats.stackexchange.com/a/581049/805 $\endgroup$
    – Glen_b
    Commented Feb 12, 2023 at 22:02

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