The school system of a major city wanted to determine the characteristics of a great

teacher, and so they asked $120$ students to rate the importance of each of the following $9$

criteria using a Likert scale of $1$ to $10$ with $10$ representing that a particular characteristic

is extremely important and $1$ representing that the characteristic is not important.

$1.$ Setting high expectations for the students

$2.$ Entertaining

$3.$ Able to communicate effectively

$4.$ Having expertise in their subject

$5.$ Able to motivate

$6.$ Caring

$7.$ Charismatic

$8.$ Having a passion for teaching

$9.$ Friendly and easy$-$going

Review Outputs $1,2,$ and $3$ in order to answer the following questions:

a$.(5$ points$)$ Based on class lecture and notes which of the $9$ variables should be included in

a Principal Components Analysis and why?

b$.(5$ points$)$ If data reduction is the purpose of the analysis. How many components should

be selected and why?

c$.(6$ points$)$ If researchers decided to use your component selection from $($b$.)$ for

interpretation, describe and evaluate the challenges that they may face and possible ways

to remedy the issue$($s$)$ identified.

d$.(6$ points$)$ Compare and discuss your results from $($b$.)$ with the output from the Variable

Clustering analysis.