Can I use inferential statistics in a non-probabilistic sample? Can I use inferential statistics (eg, binomial test, One-Sample Chi-Square Test etc) in a non-probabilistic sample (convenience sample)?
The sample size (n) = 45.
Is there any reference to support the use of inferential statistics in a non-probabilistic sample?
 A: According to Hubbard et al., (2019) the answer is YES.
Nonrandom samples, allied with a replication strategy, can yield robust findings. Which is to say that, over time, via replication research, the point estimates and confidence intervals of additional (new) results about the phenomena at hand may be compared with the increasingly sturdy bench marks established by their numerous predecessors to check for consistency. Consistency gained in this manner leads to deserved support for the veracity of the findings.
And striking break throughs in other fields have arisen from the employment of such samples. For example, Freedman (1999) tells of how the use of convenience samples was key to establishing the causal links betweencontaminatedwater andthe spreadof cholera, and cigarette smoking and lung cancer.
References
Freedman, D. (1999), “From Association to Causation: Some Remarks on the History of Statistics,” Statistical Science, 14, 243–258.
Hubbard, Raymond; Haig, Brian D.; Parsa, Rahul A.  (2019). The Limited Role of Formal Statistical Inference in Scientific Inference. The American Statistician, 73(sup1), 91–98.         doi:10.1080/00031305.2018.1464947.
A: Of course you can.  The resulting probabilities are completely meaningless, but you can certainly plug numbers into formulas.  Pretty much all NHST inference assumes randomly sampled data.
