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.

a. Which value of k appears to be the most appropriate to categorize these wines?

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?

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k Average Silhouette Value 2 0.259 0.285 0.260 5 0.231 67 0.194 0.206 8 0.197 9 0.156 10 0.202