Factor Analysis

A pilot survey was conducted of $30$ individuals to understand their different attitudes towards discount stores $($e$.$g$.,$ K$-$Mart$),$ which usually come with low levels of service versus departmental stores $($e$.$g$.,$ Macys$)$ which come with higher levels of service. Respondents were asked their opinion on a $0-9$ agree$-$disagree scale $($where $0$ indicates strongly disagree and $9$ indicates strongly agree$)$ on the following questions.

I never go back to a store that had treated me with disrespect.$($X$1)$

I find salespeople who fawn over me irritating, I just expect them to courteous.$($X$2)$

I have a lot of questions when I shop, I greatly appreciate a salesperson who is willing to spend time answering my questions.$($X$3)$

I care little for the fancy displays in departmental stores.$($X$4)$

Discount stores are much more of a bargain than departmental stores.$($X$5)$

The data for this study are attached $.$ Answer the following questions

Suppose you were looking to reduce the $5$ variable dataset into $2$ factors. Show the correlation matrix. Based on the correlation matrix of the variables, which variables do you expect to be associated with Factor $1$ and Factor $2.($Note the labels factor$1$ and factor $2$ are arbitrary$).2$ points

Do a Principal components analysis of the data. How many factors would you keep based on the Scree Plot? How much variance is explained by the number of factors you wish to keep? $3$ points

Do a factor analysis of the data without any rotations keeping $2$ factors. Can you find interpretable factors? Based on the factor loading results can you explain, why you can interpret the factors or why not? $2$ points

Do a factor analysis of the data with varimax rotation keeping $2$ factors. Interpret the two factors you discovered with varimax rotation. $2$ points

Plot respondent using the two factors. How many segments of consumers do you see in the data? On the score plot, draw circles or ovals around the segments to indicate your description of these segments. $($print plot, circle segments$/$take photo using app like tiny pdf scanner$/$ or scan the graph, add to document and upload your respons$).$ Describe the characteristics of these segments.