Question

# If 95% Tukey confidence intervals tell us to reject H_{0}:μ_{1}=μ_{2} and H_{0}:μ_{1}=μ_{3}, will we necessarily reject H_{0}:μ_{2}=μ_{3}?

Confidence intervals
If 95% Tukey confidence intervals tell us to reject $$\displaystyle{H}_{{{0}}}:μ_{{{1}}}=μ_{{{2}}}$$ and $$\displaystyle{H}_{{{0}}}:μ_{{{1}}}=μ_{{{3}}}$$, will we necessarily reject $$\displaystyle{H}_{{{0}}}:μ_{{{2}}}=μ_{{{3}}}?$$
If we have 3 groups and we calculate the Tukey confidence interval for each pair of means difference and we reject $$\displaystyle{H}_{{{0}}}:μ_{{{1}}}=μ_{{{2}}}$$ and $$\displaystyle{H}_{{{0}}}:μ_{{{1}}}=μ_{{{3}}},{W}{e}\neg\ne{c}{e}{s}{s}{a}{r}{i}{l}{y}{r}{e}{j}{e}{c}{t}{h}{y}{p}{o}{t}{h}{e}{s}{e}{s}{P}{S}{K}{H}_{{{0}}}:μ_{{{2}}}=μ_{{{3}}}$$. We had in a, previous task this situation.
For example, if $$\displaystyleμ_{{{2}}}$$ is very close to $$\displaystyleμ_{{{3}}}$$ this means that $$\displaystyleμ_{{{2}}}\approxμ_{{{3}}}\approxμ$$. So, since it would be almost the same value for both means, they will be different from ja1. But, also it can be that all three means are very different one from another.