53 54 55 56 157 58 159 ### Problem 4 continue with the workout.csv dataset. (a) Find the permutation-based cutoff for Tukey HSD. {r} alpha =0.05 R = 2000 R. perms = sapply(1:R, function(i) { 160 161 162 163 164 165 166 167 168 169 }) 170 171 172 173 174 permute.data = workout permute.data$Groups = sample (permute. data$Groups, length (permute. data$Groups), replace = FALSE) Ri round (aggregate (BPM Groups, data = permute. data, mean) $BPM, 2) all. Tij all. diff = as.numeric(dist (Ri, method = "manhattan")) max. diff = max(all. Tij) return(max. diff) tukey. cutoff tukey. cutoff = quantile (R. perms, 1-0.05) (b) Conduct \(\frac{K(K-1)}{2}\) two-sample tests (using the wilcoxon-Mann-Whitney function in R), and report their p-values. 175 {r} 176 avg_rank_group