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Table 4 H0 and HA for common scenarios. HA has also been transformed into its logical equivalent to identify HT (in bold)

From: A logical analysis of null hypothesis significance testing using popular terminology

Scenario General Form Comparing proportions (Chi-squared test) Correlation
H0 and HA H0: there is no finding in the population and the finding in the sample group is due to chance alone
HA: it is not the case that H0, therefore
HA: it is not the case that (there is no finding in the population and the finding in the sample group is due to chance alone)
[(there is no finding in the population and the finding in the sample group is not due to chance alone) or (there is a finding in the population and the finding in the sample group is not due to the population finding alone) or (there is a finding in the population and the finding in the sample group is due to the population finding alone)]
H0: (p̂1 = p̂2) [(p1 ≠ p2) due to chance alone]
HA: ¬H0, therefore
HA: ¬{(p̂1 = p̂2) [(p1 ≠ p2) due to chance alone]} ≡
HA: ({(p̂1 = p̂2) [(p1 ≠ p2) not due to chance alone]} {(p̂1 ≠ p̂2) [(p1 ≠ p2) not due to (p̂1 ≠ p̂2) alone]} {(p̂1 ≠ p̂2) [(p1 ≠ p2) due to (p̂1 ≠ p̂2) alone]})
H0: (ρ = 0) (r ≠ 0 due to chance alone)
HA: ¬H0, therefore
HA: ¬[(ρ = 0) (r ≠ 0 due to chance alone)] ≡
HA: ([(ρ = 0) (r ≠ 0 not due to chance alone)] {(ρ ≠ 0) [r ≠ 0 not due to (ρ ≠ 0) alone]} {(ρ ≠ 0) ʌ [r ≠ 0 due to (ρ ≠ 0) alone]})