Amanda Boleyn, an entrepreneur who recently sold her start-up for a multi-million-dollar sum, is looking for alternate investments for her newfound fortune. She is considering an investment in wine, similar to how some people invest in rare coins and fine art. To educate herself on the properties of fine wine, she has collected data on 13 different characteristics of 178 wines. Amanda has applied k-means clustering to this data for k = 1, ..., 10 and generated the following plot of total sums of squared deviations. After analyzing this plot, Amanda generates summaries for k = 2, 3, and 4. Which value of k is the most appropriate to categorize these wines? Justify your choice with calculations. Sum of WithinSS 500 1000 1500 2000 0 k = 2 Cluster 1 Cluster 2 Cluster 11 Cluster 2 Total k = 3 Cluster 1 Cluster 2 Cluster 3 Cluster 1 Cluster 2 Cluster 3 Total k = 4 Cluster 1 Cluster 2 Cluster 3 Cluster 4 Sum of WithinSS Over Number of Clusters +-- T 2 Cluster 1 0 5.640 Size 87 91 178 Cluster 1 0 5.147 6.078 Cluster 1 0 5.255 6.070 4.853 Size 81 62 65 51 178 4 O Inter-Cluster Distances Within-Cluster Summary Number of Clusters. Sum(WithinSS) Diff previous Sum(WithinSS) X----x-- 0 5.432 T 6 Cluster 2 5.640 0 Average Distance 3.355 3.999 3.483 3.627 Inter-Cluster Distances Cluster 2 5.147 Within-Cluster Summary Average Distance 4.003 4.260 4.134 Inter-Cluster Distances Cluster 2 Cluster 3 5.255 0 5.136 4.789 6.070 5.136 0 6.074 11 T 8 Cluster 3 6.078 5.432 0 Cluster 4 4.853 4.789 6.074 0 10 O X---- X T 10 Cluster 2 Cluster 3 Cluster 4 Cluster 1 Cluster 2 Cluster 3 Cluster 4 Total k = 2 5.255 6.070 4.853 k = 3 0 5.136 4.789 Within-Cluster Summary Average Distance Size 56 45 49 28 178 k = 4 3.024 3.490 3.426 4.580 3.498 Do not round intermediate calculations. If required, round your answers to two decimal places. 5.136 0 6.074 Cluster 1 to Cluster 2 Distance / Cluster 1 Average Distance = Cluster 2 to Cluster 1 Distance / Cluster 2 Average Distance = Average = Cluster 1 to Cluster 2 Distance / Cluster 1 Average Distance = Cluster 2 to Cluster 1 Distance / Cluster 2 Average Distance = Cluster 1 to Cluster 3 Distance / Cluster 1 Average Distance = Cluster 3 to Cluster 1 Distance / Cluster 3 Average Distance = Cluster 2 to Cluster 3 Distance / Cluster 2 Average Distance = Cluster 3 to Cluster 2 Distance / Cluster 3 Average Distance = Average = 4.789 6.074 0 Cluster 1 to Cluster 2 Distance / Cluster 1 Average Distance = Cluster 2 to Cluster 1 Distance / Cluster 2 Average Distance = Cluster 1 to Cluster 3 Distance / Cluster 1 Average Distance = Cluster 3 to Cluster 1 Distance / Cluster 3 Average Distance = Cluster 1 to Cluster 4 Distance / Cluster 1 Average Distance = Cluster 4 to Cluster 1 Distance / Cluster 4 Average Distance = Cluster 2 to Cluster 3 Distance / Cluster 2 Average Distance = Cluster 3 to Cluster 2 Distance / Cluster 3 Average Distance = Cluster 2 to Cluster 4 Distance / Cluster 2 Average Distance = Cluster 4 to Cluster 2 Distance / Cluster 4 Average Distance = Cluster 3 to Cluster 4 Distance / Cluster 3 Average Distance = Cluster 4 to Cluster 3 Distance / Cluster 4 Average Distance = Average = Based on the individual ratio values and the average ratio values for each value of k, it appears that Select your answer is the best clustering.