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QBUS1040: Foundations of Business Analytics

This assignment consists of fifteen questions in total. There are eleven written questions that you need to submit on GradeScope and fouredPencasts to be submitted via ed.

You should submit aPDF to GradeScope and match the page number with the questions you answered. You can find detailed instructions on scanning and submitting your assignments on Canvas. If you fail to match the page to the corresponding question, the marker will not be able to view your response, and thus, you will be awarded a 0 mark for the question.

You may not use notation, concepts or material from other classes (say, a linear algebra class you may have taken) in your solutions. All the problems can be done using only the material from this class, and we will deduct points from solutions that refer to outside material. You must show all of your working.

Your solutions should be neat. We will deduct points from poorly written solutions even if they are correct.

Late submissions will not be accepted unless a simple extension or special consideration was obtained.

A note about Pencast marks

The marks for each Pencast depend on the number of  exercises and quizzes, each is worth 0.5 marks. You will receive the mark if you have a next to the slide.

Question 1 (4 points) You are working with a marketing consultant who has been assigned the task of segmenting the market into distinct groups of customers for a new advertising campaign. You apply the k-means algorithm with k = 15 on a dataset consisting of 1,000,000 customer records of demographic information stored in feature vectors of length 20. The first three entries of each feature vector z, include information on the ith person's gender (1 if male, O if otherwise), the ith person's age (in years), and the ith person's annual
income (in $1,000's of dollars).
After running the k-means algorithm, you find that z12 = (0.34,31.5,87.0,...).

Give an interpretation (in words) of these results.
Question 2Recall the k-means algorithm:

given ri....,rN €R" and zq,...,zk ∈R
repeat
Update partition: assign i to Gj, j = arg minj=1,xzi-²y
Update centroids: =y=∑ieGgi
until #q....g stop changing
You are performing k-means clustering to compress the number of colours needed to rep- resent an image of The University of Sydney Quadrangle with k = 2.For simplicity, we will only consider the first 5 vectors tp....,z5, which contain RGBvalues of five different pixels, and 2 initial centroids z, and 22.
x1=(0.6，0.5，0.9)
x2(0.2,0.5，0.8)
51540
x3= (0.7，0.5，0.4)
x4= (0.4,0.4,0.7)
5=(0.3，0.4，0.2)
21 =(0.3, 0.5，0.8)
2=(0.3，0.5，0.3)
Each of the above vectors represents the RGB values of a pixel as shown below.