Origins
Hypothesis testing is largely the product of Ronald Fisher, Jerzy Neyman, Karl Pearson and (son) Egon Pearson. Fisher was an agricultural statistician who emphasized rigorous experimental design and methods to extract a result from few samples assuming Gaussian distributions. Neyman (who teamed with the younger Pearson) emphasized mathematical rigor and methods to obtain more results from many samples and a wider range of distributions. Modern hypothesis testing is an (extended) hybrid of the Fisher vs Neyman/Pearson formulation, methods and terminology developed in the early 20th century.
Fisher popularized the "significance test". He required a null-hypothesis (corresponding to a population frequency distribution) and a sample. His (now familiar) calculations determined whether to reject the null-hypothesis or not. Significance testing did not utilize an alternative hypothesis so there was no concept of a Type II error.
Neyman & Pearson considered a different problem (which they called "hypothesis testing"). They initially considered two simple hypotheses (both with frequency distributions). They calculated two probabilities and typically selected the hypothesis associated with the higher probability (the hypothesis more likely to have generated the sample). Their method always selected a hypothesis. It also allowed the calculation of both types of error probabilities.
Fisher and Neyman/Pearson clashed bitterly. The pair considered their formulation to be an improved generalization of significance testing. Fisher thought that it was without application. (The defining paper was abstract. Mathematicians have generalized and refined the theory for three generations.) All parties moved on to other matters with the conflict unresolved.
The modern version of hypothesis testing is a hybrid of the two approaches. (But signal detection, for example, still uses the Neyman/Pearson formulation.) Great conceptual differences were ignored. Neyman and Pearson provided the stronger terminology, the more rigorous mathematics and the more consistent philosophy, but the subject taught today in introductory statistics has more similarities with Fisher's method than theirs. This history explains the inconsistent terminology (example: the null hypothesis is never accepted, but there is a region of acceptance).
While hypothesis testing was popularized early in the 20th century, evidence of its use can be found much earlier. In the 1770s Laplace considered the statistics of almost half a million births. The statistics showed an excess of boys compared to girls. He concluded by calculation of a p-value that the excess was a real, but unexplained, effect.
Read more about this topic: Statistical Hypothesis Testing
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