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

H₀ and H_A

H₀: there is no finding in the population and the finding in the sample group is due to chance alone

H_A: it is not the case that H₀, therefore

H_A: 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]

H_A: ¬H₀, therefore

H_A: ¬{(p̂₁ = p̂₂) ∧ [(p₁ ≠ p₂) due to chance alone]} ≡

H_A: ({(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)

H_A: ¬H₀, therefore

H_A: ¬[(ρ = 0) ∧ (r ≠ 0 due to chance alone)] ≡

H_A: ([(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]})