# A balanced experimental design has a sample size of n=11 observations at each of k=3 factor levels. The sample averages are x¯1⋅=48.05,x¯2⋅=44.74, and

A balanced experimental design has a sample size of $$n=11$$ observations at each of $$k=3$$ factor levels. The sample averages are $$\overline{x}_{1⋅}=48.05,\overline{x}_{2⋅}=44.74, and\ \overline{x}_{3⋅}=49.11, and\ MSE=4.96$$.
(a) Calculate pairwise confidence intervals for the factor level means with an overall confidence level of 95%.
(b) Make a diagram showing which factor level means are known to be different and which ones are indistinguishable.
(c) What additional sampling would you recommend to reduce the lengths of the pairwise confidence intervals to no more than 2.0 ?

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a) $$\displaystyle\mu_{{{11}}}-\mu_{{{12}}}\in{\left({0.97},{5.65}\right)}$$
$$\mu_{11}-\mu_{13}\in (-3.40,1.28) \mu_{11}-\mu_{13}\in (-6.67,-2.03)$$
c) 50