The Boleyn sisters, a pair of entrepreneurs who recently sold their start$-$up for a multi$-$million$-$dollar sum, are looking for alternate investments for their newfound fortune. They are considering an investment in wine, similar to how some people invest in rare coins and fine art. To learn more about the properties of fine wine, they have collected data on $13$ different characteristics of $178$ wines. They have applied $k-$means clustering to these data for a range of $k$ values and generated the following table of average silhouette values and elbow chart.

$k$ Average Silhouette Value

$\backslash$table$[[2,0\mathrm{.}259],[3,0\mathrm{.}285],[4,0\mathrm{.}260],[5,0\mathrm{.}231],[6,0\mathrm{.}194],[7,0\mathrm{.}206],[8,0\mathrm{.}197],[9,0\mathrm{.}156],[10,0\mathrm{.}202]]$

$($a$)$ Which value of $k$ appears to be the most appropriate to categorize these wines?

The largest silhouette value occurs at $$ and elbow chart appears Thearly linear beyond $,$ so a set of $$

$$ clusters seems to be the most appropriate.

$($i$)$

$($b$)$ Because the average silhouette values are low for all values of $k,$ the Boleyn sisters are concerned about the quality of their clustering results. What experiments could they conduct to potentially improve the clusters?

The clustering currently is based on a relatively $$ set of variables which $$ the $$ of the clusters.

Amanda could examine the formation of clusters on a $-$ Select your answer $-$ set of variables either by $-$ Select your answer $-V$ variables from the current set or bv first performing $-$ Select your answer $-vv$ on the data and clustering on a $-$ Select your answer $-$ set of principal components.