The following figures illustrate a step in the application of the Classification Tree method to the Riding Mowers case study $(24$ total observations$).$

The first three splits are shown on the scatter plot, in which Region $1$ has $1$ observation, Region $2$ has $7$ observations, Region $3$ has $9$ observations, and Region $4$ has $7$ observations.

The tree diagram shows the classification algorithm that corresponds to the displayed three splits.

Figure $1$ Figure $2$

$1.$ Compute the Gini impurity measure for Region $2$ in the scatter plot.

$2.$ Compute the Gini impurity measure for Region $3$ in the scatter plot.

$3.$ Compute the Gini impurity measure for Region $4$ in the scatter plot.

$4.$ Compute the Combined Gini Impurity for the entire dataset at this step of the algorithm.

Consider the following Test dataset consisting of $4$ consumers and their actual ownership status:

Obs # Income Lot$\_$Size Ownership

$1110\mathrm{.}120\mathrm{.}8$ Owner

$2108\mathrm{.}017\mathrm{.}2$ Owner

$382\mathrm{.}820\mathrm{.}4$ Non

$469\mathrm{.}017\mathrm{.}6$ Owner

The algorithm shown in the tree diagram in Figure $2$ will correctly classify observation #$1$ as an Owner $($class $1)$ and incorrectly classify observation #$2$ as a Non $($class $0).$ With this as a hint to enable you to

verify your understanding of the method, use the tree diagram to classify the remaining observations. Report your answers via the next two questions.

$5.$ How will the algorithm shown in the tree diagram classify observation #$3$ from this Test dataset? $($Owner vs$.$ Non$)$

$6.$ How will the algorithm shown in the tree diagram classify observation #$4$ from this Test dataset? $($Owner vs$.$ Non$)$

With $1$ and $0$ representing "Owner" and "Non" respectively, below is the Confusion Matrix for the classification algorithm encoded in the tree diagram in Figure $2$ when applied to the entire $4-$ observation Test dataset presented above:

$7.$ What is the numerical value of B in the Confusion Matrix?

$8.$ What is the numerical value of C in the Confusion Matrix?

$9.$ Is the tree in Figure $2$ fully$-$grown?

$10.$ Would pruning the tree in Figure $2$ increase the risk of overfitting?

The following questions consider part $1$ of this module's exercises $($Classification Tree applied to the Undecideds dataset, with pruning$).$ Answer all questions for the Best Pruned tree resulting from a $50$:$30$:$20$ partition using seed$=10101.$

$11.$ Which of the predictor variables were determined by the tree to not be sufficiently relevant to the classification decision?

$12.$ Suppose Voter A has these attributes: Age$=39,$ HomeOwner$=0,$ Female$=0,$ Married$=0,$ HouseholdSize$=2,$ Income$=99,$ Education$=12,$ Church$=1.$ After the tree makes its prediction for Voter A$,$ you discover that the values for Church and Married were misrecorded. What can you say about how the tree's classification of Voter A will be affected by this discovery?