Solved Problems

A materials engineer tests the tensile strength (in MPa) of steel bars produced by three different manufacturing processes. The ANOVA results yield a Between-Group Sum of Squares ($SSB$) of 150 and a Within-Group Sum of Squares ($SSW$) of 200. There are 5 samples per process ($k=3$, total $N=15$). Conduct the ANOVA test at $\alpha = 0.05$.

A soil mechanics laboratory wants to test if the permeability of a certain soil type varies significantly depending on the compaction method used. Four different compaction methods (A, B, C, D) are tested, with 4 samples per method ($k=4$, $N=16$). The resulting permeability coefficients ($k \times 10^{-5}$ cm/s) are recorded:

Test the hypothesis that the mean permeability is the same across all four compaction methods at $\alpha = 0.01$. The sum of squares are given as: Total Sum of Squares ($SST$) = 3.52, Between-Group Sum of Squares ($SSB$) = 3.24.

Following the rejection of the null hypothesis in Problem 1, the materials engineer needs to determine exactly which of the three manufacturing processes (A, B, or C) differ from each other. The sample means are $\bar{x}_A = 400$ MPa, $\bar{x}_B = 420$ MPa, and $\bar{x}_C = 415$ MPa. What statistical procedure should the engineer use, and why is performing multiple individual t-tests inappropriate?