Question

Research conducted by Worldwide, Inc., a manufacturer of laptop computers, shows that potential laptop customers (i.e., “buyers”) differ in the importance that they attach to the following two laptop features:   (1) that the laptop include a solid state drive (i.e., SSD), and (2) that the laptop include a high resolution screen. A sample of five representative (prospective) customers of Worldwide reveals the following set of preferences for these attributes/benefits (the data are comma-delimited):

1, 6,    8

2, 3,     4

3, 4,     1

4, 4,     8

5, 5.5, 7       (this is not a typo: the x value for customer #5 is 5 ½)

where:

                  (1) the measurement scale is continuous, and ranges from 1=very unimportant to

     10=very important

                  (2) the first entry in a row is the respondent i.d. number

                  (3) the second entry (x-axis) is the importance weight attached to “includes a SSD”

                  (4) the third entry (y-axis) is the importance weight attached to “includes a high resolution

     screen”

For Question #1, parts (a) through (h) below, perform a k-means cluster analysis of the Worldwide data. For purposes of this question, set k= 2, and use the point (3,5) as initial centroid #1 and the point (6,1) as initial centroid #2. Perform all numeric calculations to 3 decimal places of precision (e.g., 8.352).

(1a) Which customers (i.e., “buyers”) are assigned to starting centroid #1, and what is the Euclidean distance between each of these customers and starting centroid #1? In your answer clearly indicate each customer’s id, and the Euclidean distance (to 3 decimal places) between the customer and starting (i.e., initial) centroid #1.

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

                  (Note: here, and below, complete for as many customers as appropriate)

(1b) Which customers are assigned to starting centroid #2, and what is the Euclidean distance between each of these customers and starting centroid #2? In your answer clearly indicate each customer’s id, and the Euclidean distance (to 3 decimal places) between the customer and starting centroid #2.

Buyer id: _________          Distance: ____________       

                                    Buyer id: _________          Distance: ____________       

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

(1c) Following your assignment (in 1a and 1b above) of customers to the two starting centroids, what are the revised (i.e., updated) centroid values for centroid #1 and centroid #2?   (Note: you can refer to these revisions as “1^st iteration”-revised centroids)

                  1^st iteration-revised centroid #1: ___________  

                  1^st iteration-revised centroid #2: ___________  

(1d) Next, based on your answer to part (1c), and continuing the k-means clustering process, which customers should be assigned to the 1^st iteration-revised centroid #1, and what is the Euclidean distance between each of these customers and the 1^st iteration-revised centroid #1?

                                    Buyer id: _________          Distance: ____________    

                                    Buyer id: _________          Distance: ____________    

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

(1e) Similarly, based on your answer to part (1c), which customers should be assigned to the 1^st iteration-revised centroid #2, and what is the Euclidean distance between each of these customers and the 1^st iteration-revised centroid #2?

Buyer id: _________          Distance: ____________   

                                    Buyer id: _________          Distance: ____________   

                                    Buyer id: _________          Distance: ____________   

                                    Buyer id: _________          Distance: ____________

                                    Buyer id: _________          Distance: ____________

(1f) Based on your customer assignments in parts (1d) and (1e), what are the 2^nd iteration-revised centroid values, for centroid #1 and centroid #2?

                  2^nd iteration-revised centroid #1: ___________    

                  2^nd iteration-revised centroid #2: ___________  

(1g) Are any additional iterations needed in this k-means clustering problem? Yes or no? Why or why not?

                  ___________________________________________________________________

(1h) What is the Euclidean distance between the final cluster centroids (i.e., the 2^nd iteration-revised centers)?

                  Distance = _______________

Answer 1

Number of clusters:   2

            Number of    within cluster   Average distance Maximum distance

           Observations   sum of squares   from centroid    from centroid

Cluster1         3               2.833            0.952            1.213

Cluster2         2               5.000            1.581            1.581

Cluster Centroids

Variable       Cluster1     Cluster2   Grand centroid

SSD              5.1667       3.5000       4.5000

HIGHRES          7.6667       2.5000       5.6000

Distances between Cluster Centroids

               Cluster1     Cluster2

Cluster1         0.0000       5.4288

Cluster2         5.4288       0.0000

Initial Cluster Centers
		Cluster
		1			2
SSD		6.00			4.00
HIGHRESO		8.00			1.00
Iteration Historya
Iteration			Change in Cluster Centers
			1					2
1			.898					1.581
2			.000					.000
a. Convergence achieved due to no or small change in cluster centers. The maximum absolute coordinate change for any center is .000. The current iteration is 2. The minimum distance between initial centers is 7.280.
Final Cluster Centers
		Cluster
		1			2
SSD		5.17			3.50
HIGHRESO		7.67			2.50
Number of Cases in each Cluster
Cluster	1			3.000
	2			2.000
Valid				5.000
Missing				.000