In R$,$ use the data set "state.x$77".$ By only using its "Population", "Income", and "Illiteracy"

columns, apply the K$-$Means Algorithm with $3,4,$ and $5$ clusters., then answer the following

questions.

$1)$ Provide all the necessary codes in R$.$

According to the outputs,

$2)$ What are the sizes of clusters in the $3-$cluster case?

$3)$ What are the sizes of clusters in the $4-$cluster case?

$4)$ What are the sizes of clusters in the $5-$cluster case?

$5)$ For the $3-$cluster case, give two examples from the cluster, which includes Alabama.

$6)$ For the $4-$cluster case, give two examples from the cluster, which includes Alaska.

$7)$ For the $5-$cluster case, give two examples from the cluster, which includes Arizona.

$8)$ Compare the overall quality of the three solutions. Which K value would you select for

this data set? Why?

$9)$ Visualize your choice in the previous question on a scatter plot $($Population on the xaxis, Income on the y$-$axis$).$ Provide your R code and the output.