Problem $4($Principal Component Analysis $-20$ points$)$:$$The data given in the file $$Big$5.$csv$$ are $5-$point Likert items taken from the Big Five Personality Test web$-$based personality assessment.$$ Techniques, such as Principal Component Analysis $($PCA$),$ can be used to determine different types of personalities.$$ There are $19,719$ subjects in the file and $50$ variable items as follows:

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E$1$ I am the life of the party.

E$2$ I don't talk a lot.

E$3$ I feel comfortable around people.

E$4$ I keep in the background.

E$5$ I start conversations.

E$6$ I have little to say.

E$7$ I talk to a lot of different people at parties.

E$8$ I don't like to draw attention to myself.

E$9$ I don't mind being the center of attention.

E$10$ I am quiet around strangers.

N$1$ I get stressed out easily.

N$2$ I am relaxed most of the time.

N$3$ I worry about things.

N$4$ I seldom feel blue.

N$5$ I am easily disturbed.

N$6$ I get upset easily.

N$7$ I change my mood a lot.

N$8$ I have frequent mood swings.

N$9$ I get irritated easily.

N$10$ I often feel blue.

A$1$ I feel little concern for others.

A$2$ I am interested in people.

A$3$ I insult people.

A$4$ I sympathize with others' feelings.

A$5$ I am not interested in other people's problems.

A$6$ I have a soft heart.

A$7$ I am not really interested in others.

A$8$ I take time out for others.

A$9$ I feel others' emotions.

A$10$ I make people feel at ease.

C$1$ I am always prepared.

C$2$ I leave my belongings around.

C$3$ I pay attention to details.

C$4$ I make a mess of things.

C$5$ I get chores done right away.

C$6$ I often forget to put things back in their proper place.

C$7$ I like order.

C$8$ I shirk my duties.

C$9$ I follow a schedule.

C$10$ I am exacting in my work.

O$1$ I have a rich vocabulary.

O$2$ I have difficulty understanding abstract ideas.

O$3$ I have a vivid imagination.

O$4$ I am not interested in abstract ideas.

O$5$ I have excellent ideas.

O$6$ I do not have a good imagination.

O$7$ I am quick to understand things.

O$8$ I use difficult words.

O$9$ I spend time reflecting on things.

O$10$ I am full of ideas.

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A$.$How many components are need to explain $100\%$ of total variation for this data?$$ How many components are determined from the scree plot?$$ What number of components would you use in the model?

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A$.$For the number of components in part A$,$ give the formula for the first component and a brief interpretation after rotating the components.$$ What names might you give for each of the components?

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A$.$What subjects' scores have the highest and lowest values for each principal component $($only include the number of components specified in part A$.$ For each of those subjects, give the principal component scores $($again only for the number of components specified in part A$).$

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A$.$Finally, run a common factor analysis on the same data. What difference, if any, do you find? Does the factor analysis change your ability to interpret the results practically?