$($a$)$ Outline the steps in the $k-$means algorithm. Clearly state the objective function being minimized by the algorithm.

$($b$)$ Compare and contrast $k-$means clustering and $k-$medolds clustering.

$($c$)$ The table below cross$-$tabulates a $k-$means clustering solution $($columns$)$ against ground$-$truth group labels $($rows$).$Describe how the Rand Index can be used to assess the agreement between theground$-$truth labels and the clustering solution found. Point out the drawbacks of the index and its extenslon.

$($d$)$ The average stlhowette width can be used to assess the quality of a clustering solution without reference to known cluster labels. Define the stlhouette for each observation and comment on how the value of the sllhowette is interpreted.