You are studying the gene expression in different human cell types. One of your collaborators uses PCA to reduce the dimensionality of the data and try to visualize the cell types in two dimensions. You graph the first two principal components and get the plot below.

a$)(8$ pts$)$ Is this plot a fair representation of the total variation between the states? Explain why or why not in $2$ sentences or less.

b$)(8$ pts$)$ Your collaborator has argued that there are five clear groups based on a k$-$means clustering, marked in the plot above. In order to assess if $k=5$ is truly the best, you plot the sum of squares for each value of $k,$ and obtain the plot below. Based on the above plot, what value of $k$ is best for clustering the data?$3.$ You are studying the gene expression in different human cell types. One of your collaborators uses PCA to reduce the dimensionality of the data and try to visualize the cell types in two dimensions. You graph the first two principal components and get the plot below