A study with three groups obtained the following Tukey HSD q-values: q₁₂ = 1.95, q₁₃ = 4.50, q₂₃ = 2.55, and q_critical = 3.05. For each pairwise comparison, state the null hypotheses, determine which group means differ significantly, and explain what these results imply for follow-up interpretation.

H0_12: μ1 = μ2 (fail to reject); H0_13: μ1 = μ3 (reject); H0_23: μ2 = μ3 (fail to reject). Conclude that group 2 differs from both 1 and 3 because group 1 and 3 differ from each other.

H0_12: μ1 = μ2 (reject); H0_13: μ1 = μ3 (reject); H0_23: μ2 = μ3 (reject). Conclude that all means differ because at least one q was above the critical value.

H0_12: μ1 = μ2 (fail to reject); H0_13: μ1 = μ3 (reject); H0_23: μ2 = μ3 (fail to reject). Conclude that only groups 1 and 3 differ significantly; report that ANOVA rejected overall and Tukey-Kramer pinpoints 1 vs 3 as the differing pair.

H0_12: μ1 = μ2 (fail to reject); H0_13: μ1 = μ3 (reject); H0_23: μ2 = μ3 (fail to reject). Conclude that ANOVA was invalid so no pairwise conclusions are warranted unless p-values are available from software.