Always know what test statistic to use

Question

Is there a systematic way to understand what test statistic to use ahead of a random variable $X$ with population of $n$ and parameters $\mu$ and $\sigma^2$ that may or may not be unknown?

For example, if $n < 30$, $\mu, \sigma^2$ are unknowns, I should use $\frac{(n-1)S^2}{\sigma^2} \sim \mathcal{X}^2$; but if $n > 30$ I should use the approximation for $\frac{\bar{X}-\mu}{s/\sqrt{n}} \sim normal(0,1)$, etc.

I never know what test to use from $t-student$, $normal$ and $\mathcal{X}^2$ for these parameters. Is there a way to always know?

Answer 1

Apart from the above mentioned link, here's another link that I think you'll find helpful: The Link

It's relevant summary is as follows:

1. When n > 30, it will follow normal distribution . Z test is carried out, when sample size is > 30.
2. With small samples and when the degrees of freedom less than 30, the variates are not normally distributed. Hence we make use of the ‘t’ distribution and ‘t’ test of significance.
3. When sample size is large ($n>50$), samples are independent and there are a minimum of five observations expected in each group or combination of groups. $\bf\chi^2$ test can be used. This test is used in the case when observed frequencies are to be tested for their fit with expected or theoretical frequencies or to test whether two factors of classification of a set of individuals presented in the form of two- way table are independent or not.

Note: Italicized part is not mentioned in the link.

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Addendum:

For $\chi^2$ test,

1. If the sample size, $n<50$, it's recommended to use Fisher’s exact test.
2. If expected frequency, $e<5$ then pooling techniques is used.