The Logic of Hypothesis Testing

Hypothesis testing is the formal framework statisticians use to evaluate claims about population parameters using sample data. Every hypothesis test begins with two competing statements: the null hypothesis (H₀) and the alternative hypothesis (H₁ or Hₐ). The null hypothesis is the default assumption — the claim that there is no effect, no difference, or no relationship. H₀ is treated as true unless the data provide sufficient evidence to reject it. The alternative hypothesis is the research claim — what the investigator expects or wants to demonstrate: that a treatment works, that groups differ, that a relationship exists. The null is never 'accepted' or 'proven true' — we either reject it (conclude there is evidence for the alternative) or fail to reject it (conclude the data are consistent with the null, but this does not prove the null is true). Directional (one-tailed) vs. non-directional (two-tailed) tests: a one-tailed alternative specifies the direction of the effect (H₁: μ > 50 or H₁: μ < 50) and concentrates all of the rejection region on one tail of the distribution. A two-tailed alternative does not specify direction (H₁: μ ≠ 50) and splits the rejection region across both tails. One-tailed tests have more statistical power to detect an effect in the specified direction but are only appropriate when there is strong prior justification for the directional claim — using one-tailed tests to improve p-values post hoc is a form of p-hacking. The test statistic is the calculated value (z, t, F, chi-square) that summarizes how far the sample result is from the null hypothesis value, expressed in units of standard error.