According to this discussion, the highest variance $X^\prime X$ matrix should correspond with the lowest variance $\beta$ which makes sense to me. But when I ran the following lines I got some curious results...
Y <- matrix( c(303.3,
467.1,
422.8,
391.6,
403.8,
373.8,
263.5,
226.8,
183.9,
208.4,
208.2,
168.8,
245.4,
160.0,
173.2), nrow=15, ncol=1, byrow=TRUE)
X <- matrix( c(1, 3, 1, 8, 1, 1,
1, 6, 3, 3, 1, 2,
1, 4, 1, 4, 1, 5,
1, 5, 1, 3, 1, 1,
1, 7, 1, 3, 3, 1,
1, 6, 1, 3, 5, 2,
1, 6, 1, 8, 5, 3,
1, 5, 1, 8, 3, 2,
1, 1, 3, 4, 2, 1,
1, 3, 5, 2, 2, 4,
1, 2, 3, 3, 1, 3,
1, 1, 4, 3, 2, 3,
1, 6, 5, 3, 1, 5,
1, 2, 1, 5, 2, 2,
1, 5, 1, 4, 2, 1), nrow=15, ncol=6, byrow=TRUE)
apply(t(X)%*%X, MARGIN=2, FUN=sd)
apply(solve(t(X)%*%X)), MARGIN=2, FUN=sd)
What I found curious is that $X^\prime X$ yields column standard deviations as follows
19.12503 94.55721 34.12575 100.84328 47.27120 43.32551
and the standard deviatoin of the columns for $(X^\prime X)^{-1}$ are
0.75477714 0.04729509 0.09832687 0.06428046 0.03173388 0.02858314
The former would suggest that the 4th column ($\beta_3$) would have the greatest precision but the latter would suggest that ($\beta_5$) is the most precise.
Did I goof somewhere? Which one should I trust more?