$1.$ In this problem, you will perform K$-$means clustering manually, with K $=2,$ on a

small example with n $=6$ observations and p $=2$ features. The observations are as

follows.

Obs. X$1$ X$2$

$164$

$252$

$343$

$412$

$511$

$600$

$($a$)$ Plot the observations.

$($b$)$ Randomly assign a cluster label to each observation as follows:

RNGkind$($sample$.$kind $=$ "default"$)$

set.seed$(2)$

labels $<-$ sample$(2,$ nrow$($x$),$ replace $=$ T$)$

Report the cluster labels for each observation.

$($c$)$ Compute the centroids for each cluster.

$($d$)$ Assign each observation to the centroid to which it is closest, in terms of Eu$-$

clidean distance. Report the cluster labels for each observation.

$($e$)$ Update the cluster assignments via K$-$Means algorithm until they stop changing.

How many iterations did it take? Hint: it won$$t take long.

$($f$)$ Update your plot from $($a$),$ by coloring the observations according to the cluster

labels obtained.